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0 and 'f/!(s) > 0; A2. for any 'lj!(s):::; e;
{J
> 0 one can find
e
> 0 such that
17(s) :::; 6 for any s E S with
66
Chapter 4
A3. for any e > 0 there exists a finite or countable subcollection g covers X (i.e., U Us:::) X) and t/1(9) ~f sup{t/J(s): s E S}:::; e.
c
S which
•E9
Let {. S -+ JR+ be a function. We say that the set S, collection of subsets :F, and the functions {, q, t/J, satisfying Conditions A1, A2, and A3, introduce the Caratheodory dimension structure or C-structure r on X and write T = (S, :F, {, f1, '1/J). If the maps f-t Us is one-to-one then the functions{, 17, and 'ljJ can be considered as being defined on the set :F and thus, the above C-structure coincides with the C-structure introduced in Section 1. In the general case one can still follow the approach, described in Chapter 1, to define the Caratheodory dimension and lower and upper CaratModory capacities generated by the C-structure. We shall briefly outline this approach. Given a set Z c X and numbers a E lR, e > 0, we define
Mc(Z,a,e) =
i~f{ Ee(s)q(s)"}, sell
where the infimum is taken over all finite or countable subcollections g c S covering Z with '1/J(fi):::; e. By Condition A3 the function Mc(Z, a, e) is correctly defined. It is non-decreasing as e decreases. Therefore, the following limit exists:
mc(Z,a)
=
limMc(Z,a,e:).
0 such that for any 0 < e :::; f there exists a finite or countable subcollection g c S covering X such that 'ljJ ( s) = e for any s E g. Given a E R. and e > 0, let us consider a set Z C X and define
Rc(Z,a,e:)
= i~f { L
e(s)1J(s)a}'
oE9
where the infimum is taken over all finite or countable subcollections g C S covering Z such that '1/J(s) = e for any s E g, According to A3', Rc(Z,a,e:) is correctly defined. We set
!:c(Z,a)
= limRc(Z,a,e), e--+0
i"c(Z,a) =lim Rc(Z,a,e:). 0 and any set Z c X, let ns put
where the infimum is taken over all finite or countable subcollections g c S covering Z for which '1/J(s) = e for all sEQ. Let us assume that the function 77 satisfies the following condition: A4. 77(s 1 ) = rt(sz) for any S1> s2 E S for which '1/J(sl) = 'ljJ(s2). One can now correctly define the function 17(e) of a real variable e by setting 7J(e) = ?J(s) if '1/J(s) =e. One can prove that, provided Condition A4 holds, the lower and upper Caratheodory capacities of sets satisfy Theorem 2.2, 2.3, and 2.4.
Let X and X' be sets endowed with C-structures r = (S,:F,~,7J,'I/J) and r' = (S, :F', ~', r/, '1/J') respectively. One can show that the lower and upper Caratheodory capacities of sets are invariants with respect to a bijective map x: X --t X' which preserves C-structures rand r' (compare to Theorem 2.5). Let (X, I') be a Lebesgue space with a probability measure I' endowed with a C-structure r = (:F,f.,1f,'I/J). Assume that any set Ua E :F is measurable. We define the Caratheodory dimension of the measure "' dimH '-'• and lower and upper Caratheodory capacities of the measure '-'• Cape'-' and Cap 0 !-', by (3.1) and (3.2) respectively. We have that
We shall now assume that the following condition holds· A5. for !-'-almost every x E X and any e > 0, if s E S and U8 3 x is a set with '1/J(s) ~ e then 1-'(U.) > 0 and f.(s) > 0. For each point x EX and a number e, 0 < e e, we chooses= s(x,e) E S 1 such that x E U, and '1/J(s) = e (this is possible in view of Condition A3 ). Once this choice is made we obtain the subcollection
s
S'
= {s(x,e) E S: x EX, 0 < e S
~:}.
Given a E R and x E X, we define now the lower and upper a-Caratheodory pointwise dimensions of I' at x by odogJI.(U•(z,e)) ( ) _lim d !!!C,,..,a x - e->O log (f.(s(x, e))77(s(x, e))")'
a log JI.(Us(z,e)) ( ) -1'd c,,..,o x = .~ ,....lo_g_,(~-(,_s.,...(x:....,-:e)..,..)ry-'-("-s-':-(x"'",-:e)-:-)-:-")
Chapter 4
68
(see (3.5)). We have that .!k-,JA,(x) ::; dc,JA,a(x) for any x E X. It is a simple exercise to prove that the conclusion of Theorems 3.1, 3.2, 3 3, 4.1, 4.2, 5.1, and 5.2 hold (Wlth obvious modifications in the formulations). We also define ..,..,
.!:!.0
( ) I' . f o:logJ,t(U.) 1m m ,JA,a x = e->0 zEU, log (~(s)T1(S) 0 )' tb(•)=
oo lim M(Z,o:,cp,U,N),
where
M(Z,o:,cp,U,N) = inf { (i
L UE(i
1
exp (-o:m(U)
+
sup ;oEX(U)
m~- cp(fk(x)))} k=O
(11.4)
C-Structures Associated with Dynamical Systems: Thermodynarmc Formalism
69
and the infimum is taken over all finite or countable collections of strings 9 c S(U) such that m(U) ;:::: N for all U E 9 and 9 covers Z (i e., the collection of sets {X(U) : U E Q} covers Z). Furthermore, the Caratheodory functions r.c(Z, a) and 'i'c(Z, a) (where Z C X and a E lR; see Section 10) depend on the cover U and are given by rc(Z,a)
= N-4oo lim R(Z,a,!p,U,N),
rc(Z,a)
= N-+oo lim R(Z,o.,!p,U,N),
where R(Z,o.,!p,U,N) = inf { g
L
exp (-aN+
UEQ
sup
~'P(fk(x)))}
(11.4
1 )
xEX(U)k=O
and the infimum is taken over all finite or countable collections of strings 9 c S(U) such that m(U) N for all U E 9 and Q covers Z. According to Section 10, given a set Z c X, the C-structure T generates the Caratheodory dimension of Z and lower and upper Caratheodory capacities of Z specified by the cover U and the map f. We denote them by Pz('P,U), CPz('P,U), and CPz('P,U) respectively. We have that (compare to (1.3) and (2.1))
=
Pz('P,U) = inf{a: mc(Z,a) = 0} =sup{ a: mo(Z, a)= oo}, CPz(I(),U) = inf{a · rc(Z, a)= 0} =sup{ a.: ro(Z,o) = oo}, CPz('P,U) = inf{o. 'i'c(Z,o.) = 0} = sup{o: 'i'c(Z,a) = oo}.
Let lUI= max{diamU,:
u, c U} be the diameter of the cover U.
Theorem 11.1. For any set Z C X the following limits
ex~st.
Pz('P) ~f lim Pz('P,U), IUI-+0
CPz('P) ~r lim CPz('P,U),
-
IUI-+0-
CPz('P) ~f lim CPz('P,U). IUI-+0
Proof. Let V be a finite open cover of X with diameter smaller than the Lebesgue number of U. One can see that each element V E V is contained in some element U(V) E U. To any string V = {V; 0 ••• Vim} E S(V) we associate the string U(V) = {U(V;.) ... U(V; .. )} E S(U). If 9 c S(V) covers a set Z c X then U(9) = {U(V) : V E Q} c S(U) also covers Z. Let 'Y = 'Y(U) = sup{I'P(x)- 'P(Y)I: x,y E U for some U E U}. One can verify using (11.4) that for every a E JR. and N > 0 M(Z,a,I{),U,N):::; M(Z,a-'"(,I{),V,N)
(11 5)
Chapter 4
70
This implies that
Pz(cp,U)- "Y :S: Pz(cp, V). Since X is compact it has finite open covers of arbitrarily small diameter. Therefore, Pz(IP,U)- "Y :S: lim Pz(IP, V). lVI -tO
If /U/ -+ 0 then -y(U) -+ 0 and hence lim Pz(IP,U) :-:; lim Pz(cp, V). lUI-tO
IVI--+0
This implies the existence of the first limit. The existence of two other limits can be proved in a similar fashion by using the inequality which is an analog of (11.5)
(11 6)
R(Z,a,cp,U,N)::; R(Z,a--y,ip, V,N). This complett>-S the proof of the theorem.
•
We call the quantities Pz(cp), CPz(ip), and CPz(fP), respectively the topological pressure and lower and upper capacity topological pressures of the function IP on the set Z (with respect to !). Sometimes more explicit notations Pz,J(cp), CPz.J(cp), and CPz.J(IP) will be used to emphasize the dependence on the map f. We emphasize that the set Z can be arbitrary and need not be compact or invariant under the map f. If f is a homeomorphism then for any set Z c X its topological pressure coincides with topological pressure on the invariant hull of Z (i.e., the set U r(Z); this follows from Theorem 11.2 below) However, this nEZ
may not be true for lower and upper capacity topological pressures (see Example 11 2 below). We formulate the basic properties of topological pressure and lower and upper capacity topological pressures. They are immediate corollaries of the definitions and Theorems 1.1 and 2.1.
Theorem 11.2. (1) P0(1P) :S: 0. (2) Pz, (IP)::; Pz.(cp) if Z1 c Z2 c x. (3) Pz(IP) =SUP;> I Pz, (cp), where = ui>lzi and (4) Iff is a homeomorphism then Pz(cp),::::: P1(z)(cp).
z
z, c X, i =
1, 2, ....
Theorem 11.3. (1) CPG(IP) :-:; 0, CP0(IP) :-:; 0. (2) CPz, (I/') :S: CPz,(IP) and CPz,(cp) :-:; CPz,(IP) if Z1 C Z2 C X. (3) CPz(IP) 2: supi>I QEz,(cp) and CPz(IP) 2: supi>l CPz,(cp), where Z = U;> 1 z, and Zi X, i = 1, 2,. . . (4) If ii: X-+ X is a homeomorphism which commutes with f (i.e., f o h = ho f} then
c
Pz(cp)
= Ph(Z)(cp o h- 1 ),
CPz(IP) = CPh(Z)(cpoh- 1 ),
CPz(IP) = CPh(Z)(cpoh- 1 )
C-Structures Associated with Dynamical Systems: Thermodynamic Formalism
71
Obviously, the functions 11 and 'lj; satisfy Condition A4 in Section 10. Therefore, by Theorems 2.2 and 11.1, we have for any Z C X that
CPz(cp)
=
lim
1 lim N log A(Z,cp,U,N),
IUI-+0 N-+oo
-
CPz(cp) = lim
(11 7)
-1
lim Nlog A(Z,cp,U,N),
IUI-+ON-+oo
where in accordance with (2.3), (11.1), and (11.3)
A(Z, cp,U, N) = inf { g
L
UeQ
exp ( sup "I:\p(fk(x)))}
(11.8)
xEX(U) k=O
and the infimum is taken over all finite or countable collections of strings 9 C S(U) such that m(U) = N for all U E g and g covers Z. We also point out the continuity property of the topological pressure and lower and upper capacity topological pressures.
Theorem 11.4. For any two continuous functions cp and 'lj; on X IPz(cp)- Pz('I/J)I ::S:
llcp -1/111. llcp- 1/JII, ::S: II'P- 1/JII
ICPz(cp)- CPz(1/J)I ::S: ICPz(cp)- CPz(1/J)I where
11·11
denotes the supremum norm in the space of continuous functions on X.
Proof. Given N > 0, we have that
~sup
N-1
L icp(f'"(x))- 1/J(fk(x))l ::S: II'P- 1/JII.
:tEX k=O
It follows that
M(Z,a:+
llcp- 1/JII,'Ij;,U,N):::; M(Z,a.,cp,U,N):::; M(Z,a:-llcp- 1/JII,'Ij;,U,N)
This implies that
Pz('lj;,U)
-II'P -1/111
::S: Pz(cp,U) ::S: Pz('I/J,U)
+ II'P- 1/JII
and concludes the proof of the first inequality The proof of the other two in• equalities is similar.
72
Chapter 4 One can easily see that
Pz(cp):::; CPz(cp):::; CPz(cp).
(11.9)
Below we will give an example where the strict inequalities occur (see Examples 11.1 and 11.2}. The situation for invariant and compact sets is different. Theorem 11.5. (1) For any f-invanant set Z C X we have CPz(cp) = CPz(cp); moreover, for any open coverU of X, we have CPz{cp,U) = CPz(cp,U). (2) For any compact invanant set Z c X we have Pz(cp) = CP z(cp) CPz(cp); moreover, for any open cover U of X, we have Pz(cp,U) CPz(cp,U) = CPz(cp,U). Proof. Let Z c X be an !-invariant set. Choose two collections of strings 9m C Sm (U) and 9n C Sn (U) which cover Z and consider
9m,n ~f {UV: U E 9m, V E 9n} C Sm+n(U). Since Z is /-invariant the collection of strings 9m,n also covers Z. We wish to estimate A(Z, cp, U, m + n) using (11.8). We have
This implies that A(Z,cp,U,m + n):::; A(Z,cp,U, m) x A(Z,cp,U,n). Let am = logA(Z,cp,U,m). Note that A(Z,cp,U,m) 2 e-mll'l'll. Therefore, infm;?:l ~ 2 -II'PII > -oo. The desired result is now a direct consequence of (11.7) and the following lemma (we leave its proof to the reader; see Lemma 1.18 in [Bo2]) Lemma. Let am, m = 1, 2,.. be a sequence of numbers satisfying infm>l l!m. > -oo and am+n :::; am + an for all m, n 2 1. Then the limit limm->oo ~ e~sts and coincides with infm?:l ~. Choose any o > Pz(cp,U). There exist N > 0 and covers Z and Q(Z,o,Q) ~r
L exp UEQ
(
-om(U)
+
g C S(U)
m(U)-1
sup
L
zEX(U)
k=O
such that
)
cp(Jk(x))
0 and consider a point x E Z Since r covers Z there exists a string U E r such that x E X(U) and N ~ m(U) < N + M. Denote by U* the substring that consists of the first N symbols of the string U. We have that N-1
sup
m(U)-1
L'P(fk(y)) ~
yEX(U•) k=O
sup yEX(U)
L
rp(fk(y)) + Mllrpll·
k=O
If fN denotes the Collection of all SUbstrings U• COnstructed above then N-1
e-"'N L U•ErN
exp
sup
L
rp(fk(y)) ~ max{l,e-"'M}eMIJ'I'Il Q(Z,a,r) < oo.
yEX(U•) k=O
By (11.7) we obtain that a> CPz(rp), and hence the desired result follow (11.9) Theorem 11.5 shows that for a compact invariant set Z the topological pre&sure and lower and upper capacity topological pressures coincide and the common value yields the classical topological pressure (see, for example, [Bo2]). It is worth pointing out that this common value is a topological invariant (i.e , Px(rp) = Px(rp a h), where h is a homeomorphism which commutes with f). This means that the pressure does not depend on the metric on X. If a set Z is neither invariant nor compact one has three, in general distinct, quantities. the topological pressure, Pz(rp), and lower and upper capacity topological pressures, CPz(rp) and CPz(rp). The latter coincide if the set Z is invariant and may not otherwise (see Example 111 below). Furthermore, they are defined by formulae (11.7) and (11.8) which are a straightforward generalization of the classical definition of the topological pressure In view of the
74
Chapter 4
variational principle, the topological pressure Pz(cp) seems more adapted to the case of non-compact sets and plays a crucial role in the thermodynamic formalism (see Appendix II).
Remarks. (1) We describe another approach to the definition of topological pressure. Let (X,p) be a compact metric space with metric p, f:X---+ X a continuous map, and cp: X ---+ lR a continuous function. Fix a number li > 0. Given n > 0 and a point x E X, define the (n, /i)-ball at x by
Bn(x,o) Put S
=X
= {y EX: p(l(x),ti(y))::; 6,
for 0::;
i::; n}.
(11.10)
x N. We define the collection of subsets
:F = {Bn(x,o): x EX, n EN} and three functions
~. 7J, 1/J: S---+
lR as follows
~(x,n) = exp ( 71(x, n)
=
sup
I:cp(fk(y))),
yeB.(z,6) k=O
exp( -n),
1/J(x, n)
= n- 1
One can directly verify that the setS, the collection of subsets :F, and functions 71, and 1/J satisfy Conditions AI, A2, A3, and A3' in Section 10 and hence determine a C-structure T = (S,:F,f;.,rJ,t/J) on X. According to Section 10, given a set Z C X, this C-structure generates the Caratheodory dimension of Z and lower and upper CaratModory capacities of Z which depend on o. We denote them by Pz('fJ,O), CPz(cp,/i), and CPz('fJ,fJ) respectively. Let U be a finite open cover of X and li(U) its Lebesgue number. It is easily seen that for every x EX, if x E X(U) for some U E S(U) then
e,
(11.11)
It follows now from Theorem 11.1 that
Pz(cp) = limPz(cp,o), 6-+0
(2) If the map f: X ---+ X is a homeomorphism we can consider the topological pressure and lower and upper capacity topological pressures for the map f as well as for the inverse map f- 1 • If Z is an invariant subset of X then for any continuous function '{J: X ---+ R,
C-Structures Associated with Dynamical Systems: Thermodynamic Formalism
75
These equalities hold no matter whether Z is compact or not but may fail to be true if Z is not invariant (see Example 11.3 below). If Z is invariant and compact then in addition we have that
and this may fail if Z is not compact (although still invariant; see Example 11.3 below). Topological Entropy We consider the special case t.p = 0 Given a set Z C X, we call the quantities
hz(f) ~ Pz(O),
Chz(f) ~ CPz(O),
Chz(J) ~rCPz(O)
respectively, the topological entropy and lower and upper capacity topological entropies of the map f on Z. We stress again that the set Z can be arbitrary and need not be compact or invariant under f. It follows from (11.9) that (1112) If the set Z is /-invariant, we have
Chz(f) = Chz(f) ~r Chz(f). By (11.7) we obtain for an invariant set Z that 1
Chz(f) = lim lim N log A(Z, O,U, N) IUI-+ON-+oo
=lim
1 lim N log A(Z,O,U,N),
(11.13)
IUI-+ON-+oo
where, in accordance with (11 8), A( Z, 0, U, N) is the smallest number of strings U of length N, for which the sets X(U) cover Z. Formula (11.13) reveals the meaning of the quantity Chz(f): it is the exponential rate of growth inN of the smallest number of strings U of length N, for which the sets X (U) cover Z. For a compact invariant set Z we have by Theorem 11.5, that hz(f) = Chz(f) = Chz(f). The topological entropy and lower and upper capacity topological entropies have properties stated in Theorems 11.2 and 11.3 (applied to t.p = 0). In particular, they are invariant under a homeomorphism of X which commutes with f. We now proceed with the inequalities (11.12). In examples below we consider the symbolic dynamical system (Ep, fl), where Ep is the space of two-sided infinite sequences on p symbols and fl is the (two-sided) shift. We recall that the cylinder set ci,. il consists of all sequences w = (jk) for which im = im, ... , it = it (see more detailed description in Appendix II below).
76
Chapter 4
Example 11.1. There eXIsts a compact non-invanant set Z C Ea for which Chz(a) < Chz(a). Proof. Let Z
nk
be a strictly increasing sequence of integers. Define the set
= {w == (wn) E l:3 :wn =
1 or 2
if n2t
:s; n < n2t+1 and n < nzt+2 for some l}.
Wn = 1,2, or 3 if n2t+1 :s;
Obviously, the set Z is compact. Consider the sequence an defined as follows: n < nu+1 and~= 3 if n2t+1 :s; n < nze+2· Set Sn = TI~=t ak· We choose the sequence nk growing so fast that
an= 2 if nu :s;
where C1 > 0 is a constant independent of l. Given m 2: 0, consider the cover Um of l:a by cylinder sets C;_.,. ;.,. . Notice that for every string U the set X(U) is a cylinder. Therefore, in accordance with (11.8), A(Z, O,Um, N) is the smallest number of cylinders of length m + N needed to cover the set Z. It follows that if nk :s; N < nk+l• then
where C2(m) > 0 is a constant independent of n. Applymg (11.13) with U = Um (and lUI-+ 0 as m-+ 0) yields Chz(u)
= log2,
Chz(a)
= log3.
The desired result follows.
•
Example 11.2. There is an invariant set Z C l:2 for which hz(u) < Chz(u). Proof. Define the sets
zk = {w = (wn)
E E2
Wn
= 1 for alllnl2: k}, z = uzk· kEZ
It is easy to see that the set Z is invariant and everywhere dense in l:2. Therefore, Chz(u) = ChE 2 (a) = log2. Given m 2: 0, consider the cover Um of E2 by cylinder sets C;_.,. ;.,.. It is easy to see that A(Zk, 0, Um, N) :s; C, where C > 0 is a constant independent of N. Therefore, Chz"' (u) = Chz.,. (u) = 0 and hence hz.,. (u) =0 for all m. This • implies that hz(u) = 0.
C-Structures Associated with Dynamical Systems: Thermodynamic Formalism
77
Note that the set Z in this example is the invariant hull of the set Zo and Chz0 (u) = Chz0 (u) = 0 while Chz(u) = log2.
Example 11.3. (1) There is an invanant (non-compact) set Z C E 2 for which hz(u) = log2 while hz(u- 1 ) = 0. (2) There is a compact (non-invanant) set Z c E 2 for which hz(u)
= Chz(u) = Chz(o-) = log2
whue
Proof. Define the sets
Z~r. = {w = (wn)
E E2:
Wn = 1 for all n::,:: k},
Z = Uz~c. kEZ
Obviously, the set Z is mvariant (but not compact) and the set Zo is compact (but not invariant). We leave it as an exercise to the reader to show that Z fulfills requirements in Statement 1 and so does Zo in Statement 2.
Remark. Let U be a finite open cover of X. Given a set Z hz(f,U) ~r Pz(O,U),
Chz(f,U) ~ CPz(O,U),
c X, the quantities Chz(J,U) ~r CPz(O,U)
are called the topological entropy and lower and upper capacity topological entropies off on Z with respect to U. By Theorem 11.5, if Z is invariant, then Q!.z(J,U} = Chz(J,U) ~f Chz(J,U) and if, in addition, Z is compact then hz(f,U) = Chz(f,U). Let V be a finite open cover of X whose diameter does not exceed the Lebesgue number of U. Applying (11.5) with rp = 0 we obtain that hx(f,U) ::; hx(J, V).
In [BGH], Blanchard, Glasner, and Host obtained a significantly stronger statement. Namely, let € be a finite Borel partition of X such that each element of { is contained in an element of the cover U. Then there eXUlts an f-invanant ergodic measure J-1 on X for which hx(f,U)::; hp(f,€). Measure-Theoretic Entropy Let p. be a Borel probability measure on X (not necessarily invariant under f). Consider a finite open cover U of X. According to Section 10, the C-structure 7 = (S, :F, f,, T/, ¢) on X, introduced by (11.1), (11.2), and (11.3), generates the Caratheodory dimension of p, and lower and upper CaratModory
Chapter 4
78
capacities of /J specified by the cover U and the map f. We denote them by P11 (1p,U), ~(IP,U), and GP11 (1p,U) respectively. We have that
= inf {Pz(!p, U) : p(Z) =
P 11 (1p, U)
1},
GP 11 (1p,U)
= j~ inf {CPz(!p,U):
p,(Z) ~ 1- o},
GP 11 (1P,U)
= j~ inf {CPz(IP,U):
p,(Z) ~ 1- o}.
(11.14)
It follows from Theorem 11.1 that there exist the limits
P11 (1p) ~r lim P,.(IP,U), IUI->0
CPiiP)
~r lim CP,.(IP,U),
-
de£.-
-
CP,.(IP)
(11.15)
IUI->0-
= lUI-tO hm CP,.(ip,U).
Given a point x EX, we set in accordance with (10.1)
_
V ~,,.,a ( X, !p, U) -
. . f 1tm 11} N->oo
alogp,(X(U))
-Na + sup
N-l
L
cp(!k(y))
yEX{U) k=O
( U) -. V c,,.,a x, !p, = 1tm sup N-+oo u
alogp(X(U)) N-l
-Na + sup
L
1p(jk(y))
yEX{U) k:O
where the infimum and supremum are taken over all strings U with x E X(U) and m(U) = N.
Proposition 11.1. If p, is a Borel probalnlity measure on X invanant under the map f and ergodic, then for every a E R and p,-almost every x E X lim 12c,."'(x,cp,U) = lim Dc,.o.(X,!{),U) =
IUI-+0
' '
IUI-+0
' '
a7(f)d ,
a- x 'P
/.1-
where h,. (f) is the measure-theoretic entropy off.
Proof. We need the following statement known as the Brin-Katok local entropy formula (see [BK]).
Lemma. For IJ.-almost every x
E X we have
. -log~J.(B,.(x,o)) _ . lim -logp,(Bn{x,o)) _ . 1 1 -1m , 1 h ,. (/) -tmrm 6->0 ..~ n 6->0 n ..... oo n where Bn(x,o)
iB
the (n,o)-ball at x (see (11.10}).
Proof of the lemma. For the sake of reader's convenience we present a simplified version of the proof in [BK] which exploits the fact that the measure
C-Structures Associated with Dynamical Systems: Thermodynamic Formalism p. is ergodic.
Fix
o>
79
0 and consider a finite meaBurable partition { with 0 ~ o. Denote by q..(x) the element of the
diam~ ~f max{diamCe- : Ce E partition
~n =~V
r
1
{V···V
rn€
containing x. Obviously, Oen (x) C Bn(x, o). By the Shannon-McMillan-Breiman theorem the following limit exists for p.-almost every x E X:
where h,..(f, ~) is the measure-theoretic entropy off with respect to{. It follows that lim lim -logi'(Bn(x,o)) ~ h (!,{) ~ h (!). 6--+0 n--+oo n ,.. ,.. We proceed now with the estimate from below. Fix c > 0. One can show that there exists a finite measurable partition (. of X satisfying (1) h,..(J, ~) ;::: h11 (J)- e; (2) Jt(o~) = 0, where{)~ denotes the boundary of the partition{. Foro>Olet U5({)
= {x EX:
the ball B(x,o) is not contained in Cdx)}.
n
Since 6 >0 U6({) = 8(. we obtain that Jt(U6({)) -+ 0 as o-+ 0. Therefore, one can choose oo > 0 such that Jt(U6(~)) ~ e for any 0 < o : =:; oo. Hence, by the Birkhoff ergodic theorem, for p.-almost every x EX there exists N 1 (x) such that for any n;::: N 1 (x), l
n-1
.
-L:xu (e)U'(x)) ::=:;e. ni=O 6
Let At= {x EX: N 1 (x) ~ £}. Clearly, the sets At are nested and exhaust X up to a set of measure zero. Therefore, there exists £0 > 1 such that I'( At) ;::: 1 - e for any l;::: lo. Fix l;::: £0 . Given a point x EX, we call the collection
(Ce(x), Ce(J(x)), ... , Ce(f"- 1 (x))) the (~,n)-name of x. If y E Bn(x,o) then for any 0 ::=:; i ::=:; n- 1 either f'(x) and J•(v) belong to the same element of~ or J•(x) E U6 ((.). Hence, if X E At and y E Bn(x, o), then the Hamming distance between ({, n)-names of x and y does not exceed e (recall that the Hamming distance is defined aB follows: ~ I;~,:-01 Jt(Ce(/i(x)}ACe(!i(y)))). Furthermore, for x E At, Bowen's ball Bn(x,c5) is contained in the set of points y whose (~,n)-names are e-close to the ({, n)-name of x. It can be shown that the total number Ln of such ({, n)-names admits the following estimate:
80
Chapter 4
where K 1 > 1 is a constant independent of x and n. We wish to estimate the measure of those points in At whose (~, n )-names have an element of the partition ~n of measure greater than exp((-h"(f,~) + 2K1 e)n) in therr Hamming .:-neighborhood. Obviously, the total number of such elements does not exceed exp ((hi'(J, ~)- 2K1e)n). Hence, the total number Qn of elements in their Hamming .:-neighborhood satisfies
By the Shannon-McMill an-Breiman theorem for JL-almost every x E X there exists N2(x) such that for any n 2: N2(x),
Let Bk = {x EX: N 2 (x) ~ k}. Clearly, the sets Bk are nested and exhaust X up to a set of measure zero. Therefore, there exists k > 1 such that JL(Bk) 2: 1-.: for any k 2: /co. Fix such a number k and consider those of the Qn elements of whose intersection with At n Bk have positive measure To estimate their total measure Sn we multiply their number by the upper bound of their measure
en
This implies that for any sufficiently small e > 0 and 5 > 0 lim -IogJL(Bn(x,o)) >h (1")-Kc:>h (1)-c:-K.: p. ,.. 2 I' 2
-
n--+oo
n
for every point x in a subset D c At n Bk of measure 2: JL(At n Bk) - Kae (here K2 > 0 and K 3 > 0 are constants independent of x, and e). Therefore,
o,
This completes the proof of the statement. We continue the proof of the proposition. Let U be a finite open cover of X and o(U) its Lebesgue number. Since o(U) -t 0 as lUI -t 0 it follows from (11.11) and the lemma that h"(f)
=
lim
lim inf -logJL(X(U)) N . -li -logJL(X(U)) =- 11m m sup , IUI-+0 N-+oo U N lUi-tON-too U
(11.16}
where the infimum and supremum are taken over all strings U for which x E X(U) and m(U) = N. Let us fix a number c: > 0. Since
0 such that jON-->co U yEX(U)
k=O
(11.17)
N-1
=
lim lim sup sup Nl IUI-->ON-->oo u
y€X(U)
L cp (!" (y)) = k=O
{ cp d~-t. Jx
The desired result follows immediately from (11.16) and {11.17). We now use Proposition 11.1 to prove the following result. Theorem 11.6. Let f be a homeomorphism of a compact metric space X and p a non-atomic Borel ergodic measure on X. Then
Proof. Set h = h,.(f) 2: 0 and a= fx cpd~-t. We first assume that a > 0. We wish to use Theorems 3.4 and 3.5 to obtain the proper lower bound for P,.(cp) and upper bound for CP,.(cp) To do so we need to find estimates of 12c,p,a(x, cp,U) and 'Dc,,.,a (x, cp, U) from below and above respectively which do not depend on
a. Fix c, 0 < c < ~. By Proposition 11.1 one can choose p-almost every x E X,
ah 'l2c,,.,a(x,cp,U) 2: - a-a
o > 0 such that for
-o;.
Note that the function g(a) = ah(a- a)- 1 - o: is decreasing. Assuming that a varies on the interval [h +a - c, h + a], we obtain that for p-almost every x E X,
12c,p,a(x,cp,U) 2: h+a- 2c. We conclude, using Theorem3.4, that Pp(cp,U) 2: h+a-2e- and hence Pp(cp,U) 2: h +a. Since this holds for every finite open cover U by (11.15) we obtain that
P,.(cp)
~
h +a.
82
Chapter 4
We now show that CP,_.(cp):::; h+ a. Fix c > 0. Let~= {Ct.·· .,Cv} be a finite measurable partition of X with jhi-'(J,{)- hj :::; c and U = {U1 , ... , Up} a finite open cover of X of diameter :::; c for which C; C U;, i = 1, . ,p. By the Birkhoff ergodic theorem for J.'-almost every x E X there exists a number N1(x) > 0 such that for any n:?: N 1 (x),
'
~I:cp(f"(y))- al S c.
(11.18)
k=O
By the Shannon-McMillan-Breiman theorem for J.t-almost every x E X there exists a number N2(x) > 0 such that for any n :::: N2(x),
~~Jog/-I(Cen(x))+h,.(f,~)l ::;c.
(11.19)
Let !:!.. be the set of points for which (11.18) and (11.19) hold. Given N > 0, consider the set I:!..N = {x E !:!.. : Nt(x) :::; Nand N2(x) :::; N}. We have that I:!..N c I:!..N+1 and !:!.. = UN>oi:!..N Therefore, given 5 > 0, one can find No > 0 for which J.'(l:!..n 0 ) :?: 1 - 5. Fix a number N :?: No and a point x E I:!..N. Let U be a string of length m(U) = N for which x E X(U). It follows from (11.18) that
I~
sup tcp(f"(y))-a,Se+/,
(11.20)
yEX(U) k=O
where 1 = -y(U). Furthermore, using (11.19) we obtain that 1-!(C{N(x))? exp(-h- 2e:)N. This implies that the number of elements of the partition ~N that have non-empty intersection with the set I::J.N does not exceed exp(h + 2e:)N. To each element C{N of the partition f.N we associate a string U of length m(U) = N for which C{N c X(U) The collection of such strings consists of at most exp(h + 2e:)N elements which comprise a cover Q of I:!..N. By (11 8) and (11.20) we obtain that
A(I::J.N,cp,U,N):::;
L
UEQ
exp ( sup tcp(f"(y))) yEX(U)k=O
:::; exp(a + h + 3e + 1)N. In view of (11.7) this means that
CPil.N(cp,U) :::;a+h+3e+"f· This implies that CP,.(cp,U) S a+h+3c+"f. Passing to the limit as diamU -t 0 yields that CP,.(r.p) :::; a+ h + 3c. It remains to note that £ can be chosen arbitrarily small to conclude that C P,. ( cp) :::; a + h In the case a:::; 0, let us consider a function 1/J = cp + C, where C is chosen such that fx'I/Jd~-t > 0. Note that P,.(,P,U) = P"(cp,U) +C and CP~-'(1/!,U) = CP"(r.p,U) +C, and the desired result follows. •
C-Structures Associated Wlth Dynamical Systems: Thermodynamic Formalism
83
As an immediate consequence of Theorem 11.6 we obtain that
h,..(f) == P,..(O) == CP,.. (0)
= CP,..(0).
These relations reveal the "dimension" nature of the notion of measuretheoretic entropy, introduced by Kolmogorov and Sinai within the framework of general measure theory. One can obtain another "dimension" interpretation of measure-theoretic entropy using Proposition 11.1. Namely,
h,..(f) = lim f1c,.. "'(x, O,U) = lim de~-' "'(x, O,U). IUI-+0
' '
IUI-+0
' '
In conclusion, we point out a remarkable application of relations (11.14) and (11.15) known as the inverse variational principle for topological pressure:
h~-'(f)+ LVJdp.==inf{Pz(rp) In particular, when rp topological entropy
p.(Z)=l}.
= 0 this gives the inverse variational principle for
hi-' (f)= inf {hz(J): p.(Z) = 1}.
This result was first established by Bowen [Bol]. Let us also point out that the requirement in Proposition 11.1 and Theorem 11.6, that f1- is ergodic, is crucial; they may not hold true otherwise.
12. Non-additive Thermodynamic Formalism Let (X, p) be a compact metric space with metric p, f: X -t X a continuous map, and r.p = { rpn: X -t lR} a sequence of continuous functions. Consider a finite open cover U of X and definE' for each n ~ 1
"fn(r.p,U) == sup{lrpn(x)- rpn(Y)I: x,y E X(U) for some U E Sn(U)}. We assume that the following property holds: lim lim 'Yn(r.p,U) n
=0
IUI-+On-+oo
(12.1)
(since 'Yn(r.p,U) 2: 0 one can show that the limit exists as IUI-t 0). We define now the collection of subsets F by (11.1) and (11.2) and three functions (., 1], 'ljJ: S(U)-+ lR as follows
€(U)
= exp(
sup rpm(U)(x) ),
TJ(U) = exp(-m(U)),
zEX(U)
.,P(U) = m(U)- 1 •
(12 2)
Chapter 4
84
One can verify using (12.1) that the collection of subsets :F and the functions 1 in Section 10 and hence determine a C-structure r = (S,:F,e, 17, 1/;) on X. The corresponding Caratheodory function mc(Z, a) (Z c X and a E R) depends on the cover U and is given by
1/,
e, and 1/; satisfy Conditions AI, A2, A3, and A3
mc(Z,a) = lim M(Z,a,ip,U,N), N-too
where M(Z,a,ip,U,N)
= inf { E exp (-am(U) + g
ueg
sup zEX(U)
V'm(U)(:v))}
and the infimum is taken over all finite or countable collections of strings g C S(U) such that m(U)? N for all U E g and g covers Z Furthermore, the Caratheodory functions r.c(Z, a) and rc(Z, a) (where Z C X and a E IR) depend on the cover U and are given by r.c(Z,a)
= N-+oo lim R(Z,a,ip,U,N),
rc(Z,a)
= N-+oo lim R(Z,a,ip,U,N),
where R(Z, a,ip,U,N) =in£ { C
E exp (-aN+
UEQ
sup xEX(U)
V'N(x))}
and the infimum is taken over all finite or countable collections of strings g c S(U) such that m(U) = N for all U E g and g covers z. According to Section 10, given a set Z C X, this C-structure generates the Caratheodory dimension of Z and the lower and upper Caratheodory capacities of Z specified by the cover U and the map I. We denote them, respectively, by Pz(rp,U), CPz(rp,U), and GPz(rp,U). Repeating arguments in the proof of Theorem 11.1, one can show that for any Z C X the following limits exist: Pz(rp) ~£ lim Pz(rp,U), IUI-+0
GPz(rp) ~ lun CE.z(rp,U), IUI-+0
CPz(rp) ~ lim CPz(rp,U). !UI-+O
We call the quantities Pz (rp), C P z (rp), and G P z (rp), respectively, the nonadditive topological pressure and non-additive lower and upper capacity topological pressures of the sequence of functions rp on the set Z (with respect to f). They were introduced by Barreira in [Bar2]. We emphasize that the set Z can be arbitrary and need not be compact or invariant under the map I.
C-Structures Associated with Dynamical Systems: Thermodynamic Formalism
85
The quantities Pz(cp), CPz(cp), and CPz(cp) have the properties stated in Theorems 11.2 and 11.3. One can check that the functions 7] and 1/J satisfy Condition A4 in Section 10. Therefore,
CPz(cp) -
CPz(cp)
1 = IUlim lim N Iog A(Z,cp,U,N), l-+0 n-+co -1 = IUI-+0 lim lim Nlog A(Z, cp,U, N), n-+co
where, in accordance with (2.3) and (12 2), A(Z,cp,U,N)==inf{Lexp( sup rpN(x))} Q
UEQ
(12.3)
"'EX(U)
and the infimum is taken over all finite or countable collections of strings g c S(U) such that m(U) = N for all U E 9 and 9 covers Z. The use of the adjective "non-additive" is due to the following observation. A sequence of functions cp = { rpn} is called additive if
rpn+m(x) = rpn(x)
+ rpm(r(x))
for any n, m 2: 1, and x E X. One can verify that the sequence cp is additive if and only if n-1
rpn(x)
=L
rp(fk(x)),
k=O
where rp is a function. It is not difficult to check that an additive sequence satisfies (12.1) if the function rp is continuous. Furthermore, in this case, for any ZcX,
Pz(cp)
= Pz(rp), CPz(cp) = CPz(rp), CPz(cp)
= CPz(rp).
In the second part of the book we will often deal with non-additive sequences of functions One can naturally associate such sequences with geometric constructions of a general type in dimension theory (including Moran-like geometric constructions; see Chapter 5) as well as with dynamical systems of hyperbolic type (including smooth expanding maps and Axiom A diffeomorphisms; see Chapter 7). The non-additive topological pressure will be used as an essential tool in computing the Hausdorff dimension of the limit sets of geometric constructions as well as of invariant sets for hyperbolic dynamical systems. We note that the sequences of functions we will deal with, being in general non-additive, often satisfy a special property which we now describe. We call a sequence of functions cp = { rpn} sub-additive if for any n, m ~ 1, and x E X,
rpn+m(x) $ rpn(x)
+ rpm(r(x))
(12 4)
The proof of the following result is similar to the proof of Theorem 11.5.
86
Chapter 4
Theorem 12.1. [Bar2] (1) If a set Z c X lS f -invariant and a sequence of functions r.p is sub-additive then for any finite open coverU of X we have CPz(cp,U) = CPz(cp,U). (2) If a set Z C X is f -invanant and compact and a sequence of functions r.p is sub-additive and sattsfies 'Pn :=:; 'Pn+l
for some K Pz(r.p,U)
+K
> 0 then for any finite open cover U of X,
1 = CPz(cp,U) = -CPz(cp,U) = n-+oo lim Nlog A(Z, cp,U, N),
where A(Z, r.p,U, N) is gwen by {12.2).
(12.5)
Appendix II
Variational Principle for Topological Pressure; Symbolic Dynamical Systems; Bowen's Equation
The mathematical foundation of the thermodynamic formalism, i.e., the formalism of equilibrium statistical physics, has been led by Ruelle [Rl] Bowen, Ruelle, and Sinai have used the thermodynamic approach to study ergodic properties of smooth hyperbolic dynamical systems (see references and discussion in [KH)). The main constituent components of the thermodynamic formalism are: (a) the topological pressure of a continuous function cp which determines the "potential of the system"; (b) the variational principle for the topolog1eal pressure, which establishes the variational property of the "free energy" of the system (which is defined as the sum of the measure-theoretic entropy and the integral of cp with respect to a probability distribution in the phase space of the system); (c) existence, uniqueness, and ergodic properties of equilibrium measures (which are extremes of the variational principle). Ruelle's version of the thermodynamic formalism is based on the classical notion of topological pressure for compact invariant sets. In this appendix we outline more general versions of the variational principle by considering topological pressure on non-compact sets and non-additive topological pressure. We use the thermodynamic formalism to describe Gibbs measures for symbolic dynamical systems. For the reader's convenience we also provide a brief description of basic notions in symbolic dynamics which are widely used in the second part of the book. One of the main manifestations of the thermodynamic formalism in dimension theory is different versions of Bowen's equation. Its roots often provide optimal estimates (and sometimes the exact value) of the dimension of an invariant set. We describe some properties of the pressure function and study roots of Bowen's equation. In the second part of the book the reader will find many applications of these results to dimension theory as well as to the theory of dynamical systems.
Variational Principle for Topological Pressure Let (X, p) be a compact metric space with metric p, f: X-+ X a continuous map, and cp: X -+ R a continuous function. Denote by !m(X) the set of all/invariant Borel ergodic measures on X. Given an /-invariant (not necessarily
87
88
Appendix II
compact) set Z C X, denote also by !m(Z) C !m(X) the set of measures I' for which ~J.(Z) = 1. For each x EX and n?: 0 we define a probability measure 1-'z,n on X by
where 5y is the a-measure supported at the point y. Denote by V(x) the set of limit measures (in the weak topology) of the sequence of measures (tJ..,,n)nEN· It is easy to see that 0 :f. V(x) C rol(X) for each x E X. Put .C(Z) = { x E Z : V(x) nrol{Z) :f. 0}. It is easy to check that .C(Z) is a Borel /-invariant set. The following statement establishes the variational principle for the topological pressure on non-compact sets. It was proved by Pesin and Pitskel' in (PP].
Theorem A2.1. Let Z function
0.
Lemma 4. Given x E Y and IJ. E V(x) n rot(Y), there exzsts a number m > 0 such that far any n > 0 ane can find N > n and a strmg U E S(U) with m(U) = N satisfying: (1) X E X(U); (2) m(U)-1
sup xEX(U)
E
Cl'(x))
~ N(h~- n we write n n'-n f..£x,n'
=
n'J.tx,n
J.t
for some subsequence n 1. For
+ ~JI.fn(x),n'-n·
This shows that if we replace the number n; by the closest integer wluch is a factor of m then the new subsequence of measures will still converge to p. Thus, we can assume that nj = mk;. Let D 1 , .•• , Dt be the non-empty elements of the partition ~ = ( V · · · V f-{m+ll( Fix {3 > 0. For each D; one can find a compact set K; C D; such that f..£(D; \ K;) ~ {3. Each element D; is contained in an element of the cover
92
Appendix II
V = U V · · · V f-(m+l)U which we denote by B;. One can find disjoint open subsets V; such that K, C V; c B,. Moreover, there exist Borel subsets V;* comprising a Borel partition of X such that Vi c V;* c B,. Given nj = mki, we denote by M,(j) the number of those s E [0, ni ), for which f8(x) E V;*, and by M;~~ the number of those s E M;(j), for which s = q (mod m). Define pi?) t,q
= M~i) /k3'· t,q
pf.?l •
= M.(j) /n J· = .!. m (P~j) t,O + · · · + p~i) ,,m-1 ) I
Since J-Lz,ni converges to the measure J-L we obtain that hm ppl ~ J-L(K;) ~ J.L(D;) - (3, j-+oo
If j is sufficiently large and (3 is sufficiently small we find that
! (- ¥~j) logp~il)
~ (- ~J.L(D;) log J-L(Di)) + ~ ::; h,.(f) +e.
::;
Since the function g(x) = -x log x is convex we obtain that m-1
g(pJil)
~ ~ 2:;uCP~~JJ q=O
and hence
Therefore, the inequality
L9(Pl:~)::; i
L9CPlj)) i
should hold for some q E [0, m). This implies that
Let N = ni + q. For some sufficiently large j we choose a string U E S(U) with m(U) = N in the following way. For s < q we choose u. E U which contains f"(x) Further, for every V;* we choose a string U; = Uo,i .. Um-l,i such that
93
Variational Principle for Topological Pressure
Then, for s ;::: q we write s = q + mp + e with p ;=:: 0 and m > e ;::: 0 and set u. = Ue,i• where i is chosen such that r+mP(x) E Vi"· Set ap = Uo,;Ul,i ... Um-l,i , ak;-1), the and consider the string U0 ... Uq-laoa 1 ... ak,-l· For !! = (ao, measure 11-1}_ is given by probabilities vi~~, i = 1 ... t and it satisfies
This proves the first and the third statements of the lemma. Since 11-z,n; converges to the measure 11- we obtain for sufficiently large N that
This implies the second statement and completes the proof of the lemma. Given a number m > 0, denote by Ym the set of points y E Y for which Lemma 4 holds for this m and some measure J.l. E V(x) n !m(Y) We have that Y = Um:>O Ym. Denote also by Ym,u the set of points y E Ym for which Lemma 3 holds for some measure 11- E V(x) n !m(Y) satisfying Jy cpdiJ, E [u- e, u + e]. Set c = sup (h~'(f) + f cpdl-£).
}y
JSE!m(Y)
Note that if x E Ym,u then the corresponding measure 11- satisfies h,_.(j) :s; c-u+e. Let Qm,u be the collection of all strings U described in Lemma 4 that correspond to all x E Ym,u and all N exceeding some number N 0 • It follows from (A2.13) that for any x E Ym,u the substring constructed in Lemma 3 is contained in R(k,m(h + e),Um), where h = c- u +e. Therefore, the total number of the strings constructed inLemma4 does not exceed b(N) = IUimiR(k, m(h+e),Um)l. By Lemma 3 we obtain that -. Iogb(N) 1rm N
N-too
h
(A2.14)
:s; +e.
Since the collection of strings Qm,u covers the set Ym,u we conclude using Lemma 4 and (A2.14) that (
oo
M(Ym,u,>..,cp,U,No):S
L
b(N)exp
N=No
:s;
)
m(U)-1
->..m(U)+ sup zEX(U)
L
lp(jk(x))
lo=O
N~o b(N) exp ( ->.m(U) + N ([ cpdl-£ + 'Y(U) +e)) .
If N 0 is sufficiently large, we have that b(N)
:s; exp(N(h + 2e)). Hence, (3No
M(Ym,u,>., cp,U,No) :S:: l - (3'
(A2.15)
Appendix II
94
where {3 = exp
(-A+ h+ [
!pdJL+'Y(U)
+3e).
It follows from (A2.15) that if>.> c + ')'(U} + 4e then mc(Ym,u, >.) = 0. Hence, >.;:::: Py,,,.(~P,U). Assume that points U1, ... ,ur form an e-net of the interval HI~PII, II~PI!]. Then m=li=l
We have that>. 2:: Py"'·"• (~P,U) for any m and i. Therefore,
). 2::
SUJ?PYm,u; (~P,U)
m,•
= Py(~P,U)
This implies that c + 'Y(U) + 4e;:::: Py(~P,U). Since e can be chosen arbitrarily small it follows that c + 'Y(U) ;:::: Py(~P,U). Taking the limit as lUI -+ 0 yields • c ;:=:: Py (-+ h,_.(f) is upper semi-continuous on !.Dt(Z). Then the map /.1. >-+ h,_.(f) + f 1pd~ is also upper semi-continuous on !m(Z) and the desired result follows from Theorem A2.1 and the fact that an upper semicontinuous function on a compact set attains its supremum. Fix~ E !.Dt(Z), a> 0, and a partition t; = {C1 .•. Cn} of X with diamt;, ~e. For a sufficiently large m we have that
!_H,..(~ V · · · V rm+l~) ~ h,_.(f) +a.
m
Let us fix such an m. Given {3, choose compact sets
nrk(C;.)
m-1
K; 0
;.,._,
c
k=O
such that
This implies that L; ~£
m-1
U UJi(K;o
in.-1) C C;.
]=Oi;-i
Since L; are disjoint and compact one can find a partition with diam £;,' ~ e such that Li c intC;. We have that
~'
= {C~ . .
n
m-1
K; 0
;"'_ 1
c int
rk(c:.).
k=O
If a measure
and
11
E !.Dt(Z) is close to J1. in the weak•-topology then
C~}
of X
99
Variational Principle for Topological Pressure If {3 is sufficiently small it follows that
This completes the proof of the theorem. As a direct consequence of Theorem A2.2 we obtain the following statement. Theorem A2.3. Assume that a map f satisfies Conditions 1 and 2 of Theorem A2.2. Then for any compact f -invanant set Z C X and any continuous function rp on Z there exists an equilibrium measure Jl,
1, we endow the space Et with the metric I
_
00 """
1·~k - ~k., I
d{J(w,w)- ~ ~· k=O
(A2.18)
where w = (i 0 i 1 ... ) and w' = (i~ii ... ). It induces the topology on ~t such that the space is compact and cylinders are disjoint open (as well as closed) subsets. The (one-sided) shift on ~t is defined by
(we also use more explicit notation a+). It is easily seen to be continuous. A subset Q c ~t is said to be a-invariant if u(Q) = Q. When the set Q c ~t is compact and a-invariant, the map uiQ is called a (one-sided) subshift
101
Variational Principle for Topological Pressure Let A be a p x p matrix whose entries and O"-invariant subset
a;j
are either 0 or 1. The compact
is called a topological Markov chain with the transfer matrix A. The map 0"1~1 is called a (one-sided) subshift of finite type. It IS topologically transitive (i.e., for any two open subsets U, V c ~1 there exists n > 0 such that O"n(U) n V #= 0) if the matrix A is irreducible, i.e., for each entry O..i there exists a positive integer k such that a~j > 0, where a~i is the (i,j)-entry of the matrix Ak The map O"~~~ is topologically mixing (i.e., for any two open subsets U, V C ~~ there exists N > 0 such that O"n(U) n V #= 0 for any n > N) if the transfer matrix A is transitive, i.e., Ak > 0 for some positive integer k. We call (Q, O") a so fie system if Q C ~t is a finite factor of some topological Markov chain ~1, i.e , there exists a continuow; surjective map (: ~~ -+ Q such that O"IQ o ( = ( oO". An example is the even system, i.e., the set Q of sequences of 1's and 2's, where the 2's are separated by an even number of 1's Similarly to the above, we consider the space of left-sided infinite sequences
We also write w- for points in~;. A cylinder in~; is denoted by Cs_. more exphcitly C;:_,. "'). The (one-sided) shift is defined by
io
(or
(we also use more explicit notation O"-). It is continuous. Further, given a transfer matrix A= (a;j), we set ~A=
The map
0"1~:4
{w =
( ... Ltio) E ~;: a._.;_,.H = 1 for all
n EN}.
is a (one-sided) subshift of finite type.
We also consider the space of two-sided infinite sequences of p symbols
A cylinder (or a cylinder set) is defined as
C;m ;,. = where m
: 1, we endow the space E, with the metric (A2.18')
102
Appendix II
where w = (... Ltioi 1 ... ) and w' = (... i~ 1 i~ii ... ). It induces the compact topology on Ep with cylinders to be disjoint open (and at the same time closed) subsets. The (two-sided)shift u: Ep -t Ep is defined by u(w)k = WTo+l· It is an expansive homeomorphism. Given a compact u-invariant set Q C E,, we call the map uiQ a (two-sided) subshift. Let A be a p x p transfer matrix with entries 0 and 1. Consider the compact u-invariant subset
EA ={wEE,:
awnWn+l
=
1 for all n E Z}.
The map uiEA is called a (two-sided) subshift of finite type. Let us notice that given a point w E EA, the set of points w' E EA having the same past as w (i.e., w; = w~ for i ~ 0) can be identified with the cylinder c E~. Similarly, the set of points w' E EA having the same future as w (i.e., W; = w~ for i ~ 0) can be identified with the cylinder c;;, c E::i. Thus, the cylinder C;o c EA can be identified with the direct product C;~ X c;;,. For symbolic dynamical systems the definitions of topological pressure and lower and upper capacity topological pressure can be simplified based on the following observation (which we have already used in the proof of Proposition A2.1). Let Un be the open cover of Et by cylinder sets C; 0 ; ... Notice that IUnl -t 0 as n ~ oo and for any U E S(Un) the set X(U) is a cylinder set. Therefore, the function M(Z, a, cp,Un, N) can be rewritten according to (11 4) as M(Z,et,({),Un,N) =
c,:;;
i~f {
0; 0
2: im
exp (-a(m + 1) eg
+w
~up .
E •o
•m
(A2.19)
f:({)(uk(w)})} k=O
and the infimum is taken over all finite or countable collections of cylinder sets C; 0 im with m ;::: N > n which cover Z. Furthermore, the function R(Z, Ct, ({),Un, N) (N > n) can be rewritten according to (11.4') as R(Z, et, cp,Un, N)
c,o
~eQ exp (-a(N + 1) + we~~p •N~cp(uk(w}))
and the sum is taken over the collection of all cylinder sets C;0
z.
iN
(A2.19')
intersecting
Let (Q,u) be a symbolic dynamical system, where Q is a compact u-invariant subset of and cp a continuous function on Q A Borel probability measure J.1. = J.l.'P on Q is called a Gibbs measure (corresponding to ~P) if there exist constants D 1 > 0 and D2 > 0 such that for any n > 0, any cylinder set C; 0 .i .. , and any w E C; 0 ; .. we have that
Et
(A2.20)
Variational Principle for Topological Pressure
103
where P = Pq(cp). Note that if Condition (A2.20) holds for some number P then P = Pq. Indeed, in this case for every e > 0 by (A2.19) we obtain that 1
M(Z,P+e,cp,Un,N) S D1 i~f
~ 0, 0
p,(Cio i,.)exp(-e+r(Un))
imE9
S D1 1 exp( -e + r(Un)). Letting n -+ oo yields that Pq S P + e and hence Pq S P since e is arbitrary. The opposite inequality can be proved in a similar fashion (I thank S. Ferleger for pointing out this argument to me). Any Gibbs measure is an equilibrium measure but not otherwise. It is known that the spectjication property (see (KH] for definition) of a topologically mixing symbolic dynamical system (Q, u) ensures that any equilibrium measure corresponding to a Holder continuous function is Gibbs. It is known that any subshift of finite type (Et u) satisfies the specification property. Therefore, an equilibrium measure f.lcp, corresponding to a Holder continuous function cp, is a Gibbs measure provided the transfer matrix A is transitive. In this case it is also a Bernoulli measure. For an arbitrary transfer matrix A, by the Perron-Frobenius theorem one can decompose the set E~ into two shift-invariant subsets: the wandering set QI (correspondmg to the nonrecurrent states) and the non-wandering set Q2 (corresponding to the recurrent states). The latter can be further partitioned into finitely many shift-invariant subsets of the form E1,, where each matrix A; is irreducible and corresponds to a class of equivalent recurrent states (see [KHJ for details). Moreover, for each i there exists a number n; such that the map un• is topologically nuxing. Note also that any sofic system satisfies the specification property. We define the notion of Gibbs measures for two-sided subshifts. Let Q be a compact u-invariant subset of Ep and cpa continuous function on Q. A Borel probability measure p, = f.lcp on Q is called a Gibbs measure (corresponding to cp) if there exist constants D 1 > 0 and D 2 > 0 such that for any m < 0, n > 0, any cylinder set C;,. in, and any w E C;"' in we have that (A2.20')
where P = Pq(cp). Again, any Gibbs measure is an equilibrium measure and the specification property ensures otherwise. In the case of subshifts of finite type there is a deep connection between Gibbs measures for one-sided and two-sided subshifts. In order to describe this connection consider a two-sided subshift of finite type (EA, u) and a Holder continuous function
= i fori= 1, . . ,p and set n = (w< 1>, ... ,w(w) = (n/2]
:::; 2C
((i:J
pn-j
j=O
This completes the proof of the lemma.
+
L j>(n/2]
w)
(c:+-0 ' p.(ul(Ct
in
)
;J
= lim log IL n-+oo
(C
)
io in f.L(Ci 1 in)
{with uniform convergence}, and
for each w- = ( ... Ltio) E EA: (with uniform convergence). Proof. The statement is an immediate corollary of the following property of Gibbs measures (see Proposition 3.2 in (PaPo]). let f.L be the Gibbs measure corresponding to a Holder continuous function
there exists g eo m et ri c .. . ,j n = in (i co n st ru ct io n s. T he li m Such constructions are ca o h .. . ) E Q it se t F is de lled sy m b o li fined by c
=
n 00
F=
n= O
u
{so i, ) Q ·a dm iss ib
le O ne ca n clas si fy ge om et sentation. A ric constructi ons accordin construction g to their sy is called a si modeled by mbolic reprem p le g eo m th e full shift, et ri c co n st i.e., Q = E t of geometric ru ct io n if it (see Figure 1) construction is s are construc . Namely, th e tions modeled Another im po rt an t class geometric co nstruction is by subshifts ti o n if Q = E of finite type called a M ar 1 (we remind th e re . k o v g eo m et admissible w ader th at E~ ri c co n st ru it h respect to co ns cists of all sequ th e transfer m A (i j, i; + l) = ences (ioi1 .. at ri 1 .) Finally, a ge for j = 0, 1 .. . ; see Appen x A with entries A( i, j) "" 0 or 1, i.e., ometric cons di x II ; se e also Figure truction is ca (Q, o ) is a so 2 in Section lled a so fi c fic sy st em (i 13). g eo .e., a finite fa ctor o f a subs m et ri c co n st ru ct io n if hift o f finite type).
F ig u re 1. A
S IM P LE G E O M E TR
IC C O N ST
R U C TI O N . G eo m et ry o f th e co n st ru ct io n . Thi ment of th e s includes th basic sets, th e informatio ei r geometri control over n on th e plac c shapes, an th e sh ap e an ed "sizes". If d sizes of ba one has st ro sic sets, th en ng th e placemen t can be fair ly
Dimension of Cantor-like Sets and Symbolic Dynamics
119
arbitrary, and vice-versa. To illustrate this we first consider constructions whose geometry seems to be most simple where one has complete control over the shape and sizes of basic sets which are balls. Denote by r; 0 ; . the radius and by X; 0 in E ntm the center of the basic set (ball) .6.,0 in on the nth step of the construction. The collection of numbers r; 0 •· and coordinates of points Xio ;. provide complete information on the geometry of the construction and, in particular, is sufficient to compute the Hausdorff dimension and box dimension of the limit set. Surprisingly, one can often use only the numbers r; 0 in to obtain refined estimates for the Hausdorff dimension (from below) and the upper box dimension (from above) of the limit set (see Section 15) The main tool to study this geometric construction is the thermodynamic formalism, developed in Chapter 4 and Appendix II (in particular its non-additive version), applied to the underlying symbolic dynamical system. Namely, consider the sequence of functions 'Pn(w) = logr; 0 in on Q, where w = (ioii ... ). Clearly it satisfies Condition (12.1). According to Appendix II (see Theorem A2.6) the equations Pq(s{cpn}) = 0, CPq(s{cpn}) = 0 (where Pq and CPq denote the non-additive topological pressure and nonadditive upper capacity topological pressure specified by the sequence of functions {'Pn}) have unique roots provided the sequence of radii admits the asymptotic estimate (A2.23). We denote these roots by!!. and 8 respectively. Clearly, !!. :::::; 8. In Section 15 we will show that, under some mild additional assumptions, the number !!. provides a lower bound for the Hausdorff dimension and the number 8 provides an upper bound for the upper box dimension of the linrit set (see Theorem 15.1). We emphasize that the above approach works well for an arbitrary collection of numbers rio in which may depend on the whole "past" and may not admit any asymptotic behavior as n -t oo. Yet, the Hausdorff dimension and lower and upper box dimensions may not coincide and their exact values may depend on the placement of basic sets (see Example 15.1). However, if, for instance, the sequence of functions 'Pn is sub-additive (see Condition (12.4)), then the Hausdorff dimension and lower and upper box dimensions coincide and are completely determined by the sizes of basic sets only, i.e., the numbers r; 0 in. In this case !1. = s ~r sand regardless of the placement of basic sets, the Hausdorff dimension and lower and upper box dimensions of the limit set coincide and are equal to s. Moreover, s is the unique root of the equation
PQ(s{ cp,.})
= 0.
An equation of this type was first discovered by Bowen in his study of the Hausdorff dimension of quasi-circles (see [Bo2]) and now bears his name. Bowen's equation is one of the main manifestations of the general Caratheodory construction developed in Chapter 1: it expresses intimate relations between two Caratheodory dimension characteristics: the Hausdorff dimension of the linrit set of a geometric construction and non-additive topological pressure specified by the sequence of functions { 'Pn} for the underlying symbolic dynamical system
120
Chapter 5
(Q, u) Bowen's equation contains all necessary information on the geometric construction that matters in computing the Hausdorff dimension of the limit set: the information on the underlying symbolic dynamical system and its realization into the Euclidean space by basic sets Bowen's equation seems to be "universal" in dimension theory: we will demonstrate that the Hausdorff dimension of limit sets for various classes of geometric constructions as well as invariant sets for various classes of dynamical systems can be computed as roots of Bowen equations written with respect to the underlying symbolic dynamical systems. In 1946 Moran, in his seminal paper (Mo], introduced and studied geometric constructions with basic sets satisfying the following conditions· (CMl) each basic set t:-; 0 ; .. is the closure of its interior; (CM2) t:-;0 ; .. ; C t:-; 0 in for j = 1, .. ,p; (CM3) each basic set t:-; 0 ini is geometrically similar to the basic set. t:-; 0 in for every j; (CM4) for any (io ... in) =/= (jo . -Jn) the basic sets t:-; 0 ; . and t:-;0 1.. do not overlap, i.e., their interiors are disjoint; (CM5) diamt:-; 0 ini = A;diamt:-; 0 ;,., where 0 < A; < 1 for j = 1, ... ,pare constants. Such constructions are called Moran geometric constructions (the name coined by Cawley and Mauldin (CM]). The numbers A; arc known as ratio coefficients of the construction since they determine the rate of decreasing the sizes of basic sets We stress that basic sets of a Moran geometric construction at the same step may not be disjoint (they may intersect along boundaries, compare to Condition (CG3)). Therefore, the limit set F may not be a Cantor-like set and the coding map x may not be injective (but it is still surjective). Moran considered only geometric constructions modeled by the full shift. His remarkable observ-ation was that an "optimal" cover, which can be used to obtain the exact values of the Hausdorff dimension of the limit set (we call it a Moran cover) is completely determined by the symbolic representation of the geometric construction. In particular, regardless of the placement of basic sets, the Hausdorff dimension of the limit set is completely determined by the ratio coefficients. Namely, Moran discovered that it is a unique root of the equation
wh:tch is a particular case of Bowen's equation corresponding to the full shift. Moran constructions modeled by subshifts of finite type or sofic systems were studied in (MW]. In [PWl], Pesin and Weiss demonstrated that Moran's approach can be greatly extended to a much more general class of Moran-like geometric constructions with stationary (constant) ratio coefficients. They discovered that the only property of basic sets that matters in constructing Moran covers, is
Dimension of Cantor-like Sets and Symbolic Dynamics the following. there exist closed balls B;0 in of radii rio is a constant) with disjoint interiors such that
121 'n
=
0
IJj=O >.; 0 > 0 3
(
The topology and geometry of basic sets of these constructions may be quite complicated (for example, they may not be connected and their boundary may be fractal). Moreover, the basic sets at step n of the construction need not be geometrically similar to the basic sets at step n - 1 (see Condition (CM3) in the definition of Moran geometric constructions). Another important feature is that basic sets ~;0 in at step n of the construction may intersect each other (although the interiors of the balls Bio ;. must be disjoint). In [PWI], Pesin and WeiSs also showed that a slight modification of Moran's approach allows one to deal with geometric constructions modeled by an arbitrary symbolic dynamical system (Q,u). The main tool of study, as explained above, is the thermodynamic formalism described in Chapter 4 and Appendix II. In particular, regardless of the placement of basic sets the Hausdorff dimension and lower and upper box dimensions of the limit set F coincide and the common value is a unique root of Bowen's equation PQ(S.; dist(x, y) for any x, y E D (where D is the unit ball in lR.m) with fixed 0 < >.; < 1 (for detailed description of self-similar constructions and related results see [Fl] where further references can also be found). These constructions are a. very special type of Moran geometric constructions (CM1-CM5) where not only the sizes of basic sets but also the gaps between them are strongly controlled. A more general class of geometric constructions is formed by geometric constructions with contraction maps where the maps h1 are hi-Lipschitz
Dimension of Cantor-like Sets and Symbolic Dynamics
123
contraction maps. This means that for any x,y ED, _d;dist(x,y):::; dist(h;(x),h;(y)):::; :\.dist(x,y), where 0
0, and K2 > 0 are constants; (CPW4) intll.;o in nintll.;o J= = 0 for any (io . . in) =f: Uo •... ,jm) and m?: n. This class of geometric constructions was introduced by Pesin and Weiss in [PWl]. Basic sets of these constructions are essentially balls, although their topology and geometry may be quite complicated. Furthermore, basic sets Ll; 0 in at step n of the construction may intersect each other. The numbers .>.; are called ratio coefficients. They are fixed and do not depend on the basic sets. Self-similar constructions (see below) or more general Moran geometric constructions (CM1-CM5) (see the introduction to this chapter) are particular examples of geometric constructions (CPW1-CPW4). The geometric simplicity of the geometric constructions {CPW1-GPW4) will allow us to illustrate better the role of symbolic dynamics. Given a p-tuple A = (>.1> ... , Ap) such that 0 < A; < 1, there exists a uniquely defined nonnegative numberS>. such that
(we remind the reader that PQ denotes the topological pressure on Q with respect to the shift u; see Section 11 and Appendix II). Denote by x: Q --t F the coding map (see definition in the introduction to this chapter). Note that it is Holder continuous. To see this let w1 = (ioit inj . ) and w2 = (ioiJ ... i,.k ... ) be two points in Q with j =f: k. We have n
llx{wl)- x(w2)11 5
II Ai; :::; (1~ >.;)" 5 ((Jn)'":::; Cdi3(wl, w2)'"'. j=O _p
Chapter 5
124
Et
where C > 0, 0 < a < 1 are constants and d13 is the metric in (see (A2 18) in Appendix II) This implies that any Holder continuous function on F pulls back by x to a HOlder contmuous function on Q. Notice that, in general, the coding map X is not invertible (since basic sets at the same level of the construction may intersect each other) and even if it is invertible it is not necessarily Holder continuous (this depends on the placement of the basic sets, i.e , the gaps between them). Let 1-'>. be an equilibrium measure for the function (ioi 1 ... ) H B>.log >.; 0 on Q, and m>. the push forward measure to F under x (i.e., m>.(Z) = ll.>.(x- 1 (Z)) for any Borel set Z C F). We describe a special cover of the limit set F for a symbolic geometric construction (CPW1-CPW4) whlch allows one to build an opttmal cover to be used to compute the Hausdorff dimension and box dimension of F. We call this cover a Moran cover. Let >. = (>.h ... ,>.,) be a vector of numbers with 0 < >.; < 1, i = 1, ... , p. Fix 0 < r < 1. Given a point w = (~ 0 i 1 .. ) E Q, let n(w) = n(w, r, >.)denote the unique positive integer such that (13.2)
It is easy to see that n(w) ~ oo as r ~ 0 uniformly in w. Fix wE Q and consider the cylinder set C; 0 inl"'l C Q. We have that w E C; 0 . 1n 1.,.,. FUrthermore, if w' E C; 0 ;n 1..,, and n(w') ~ n(w) then Let C(w) be the largest cylinder set containing w with the property that C(w) = Cio in.). The sets ~(j) = x(C(j)), j = 1, ... , Nr are not necessarily disjoint and comprise a cover ofF (which we will denote by the same symbolilr if it does not cause any confusion). We have that ~(i) = ~io ;n.). It consists of sets c 0, there exists a number M > 0 which does not depend on x, r, and>. and satisfies the following property: the number of basic sets~ (j) in a Moran cover Ur,Q(>.) that have non-empty intersection with the ball B(x, r) 'IS bounded from above by M. We call M a Moran multiplicity factor. Moreover, given a subset R c Q, a point x E F, and a number r > 0, the number of basic sets ~(;) in a Moran cover ilr,R(>.) that have non-empty intersection with the ball B ( x, r) is bounded from above by M. )
Dimension of Cantor-like Sets and Symbolic Dynamics
125
Theorem 13.1. [PW1] Let F be the limit set for a geometnc construction (CPW1-CPW4) modeled by a symbolic dynamical system (Q,u). Then (1) dimH F = dimBF (2) dimH ffi>. = S>.; (3)
= dimBF == S>.;
Proof. Set s = S>. and d = dimH F. We first show that s ~ d. Fix e > 0. By the definition of Hausdorff dimension there exists a number r > 0 and a cover of F by balls Bt, l = 1, 2, ... of radius r~ ~ r such that (13.3)
For every l > 0 consider a Moran cover .U.., of F and choose those basic sets from the cover that intersect Be. Denote them by ~~l),.. , ~~m(l)). Note that ~~j) = ~; 0 • in 0 is a Moran multiplicity factor (which is independent of l) The sets {~~l, j = 1, .. , m(l), l = 1, 2, ... } comprise a cover Q ofF, and the corresponding cylinder sets c}'l = c,D in(l j) comprise a co~·er of Q. By (13.3) and (13.4)
Given a number N > 0, chooser so small that n(l,j) 2: N for alll and j. We now have that for any n > 0 and N > n,
M(Q,O,cp,Un,N) ~
L 6~;lEg
exp
Chapter 5
126
where M(Q,O,tp,Un,N) is defined by (A2.19) (see Appendix II) with
tp(w) = (d +e) log A;0 (and a= 0). This implies that Pq((d +e) logA;0 ) $0.
Hence, by Theorem A2.5 (see Appendix II), 8 $ d+e. Since this inequality holds for all e we conclude that 8 $ d. Denoted= dimBF. We now show that d $ s. Fixe> 0. By the definition of the upper box dimension (see Section 6) there exists a number r = r(e) > 0 such that N(F,r) 2': r•-il (recall that N(F,r) is the least number of balls of radius r needed to cover the set F). Consider a Moran cover U,. of Q by basic where · Let ~(il = x(C(;l) = ~·IO ··'n.(:x;)' j = 1 . N.. · sets C(i) =C.1-0 -ln(wJ) r x; = x(wj) Note that this cover need not be optimal, i.e., Nr ;::: N(F,r). By (13.2) there exists A > 1 such that for j = 1,.. , Nr, '
1
'
n(w;)+l
< ~ A-
IJ
,x.,, < - r
k=O
and hence
A
l
C2log- -1$ n(w;) $ C3log- + 1, r r that n(w;) can take on at implies This constants. are 0 > C and 0 > C where 2 3 C2log : + 2 possible values. most B ~r C3log We now think of having Nr balls and B baskets. Then there exists a basket containing at least 1ft balls. This implies that there exists a positive integer N E [C2log ~ - 1, C3log + 1] such that
1-
1
card {j: n(w;)
= N} 2:
N(F r) Br 2: - B ' 2:
N
r•-J A,
C3logr
where card denotes the cardinality of the corresponding set. If r is sufficiently small we obtain that card {j : n(w;) = N}
2'-_
r 2•-d.
Consider an arbitrary cover g of Q by cylinder sets C;0
;N.
It follows that
Dimension of Cantor-like Sets and Symbolic Dynamics
127
where C4 > 0 is a constant. We now have that for any n > 0 and N > n,
where R( Q. 0, cp, U,., N) is defined by (A2.19') (see Appendix II) with a = 0 and
cp(w)
= (d- 2c-) log.-\io·
By Theorem 11.5 this implies that
and hence d- 2£ :.::; s (see Theorem A2.5 in Appendix II). Since this inequality holds for all £ we conclude that d :.::; s. This completes the proof of the first statement. In order to prove the second statement we need only to establish that s :.::; dimn m;.. Assume first that the measure J-L>. is a Gibbs measure corresponding to the function slog >.io. By (A2.20) (see Appendix II) there exist positive constants D 1 and D2 such that for J = 1, ... , Nr (13.5) Consider the open Euclidean ball B(x, r) of radius r centered at a point x. Let N(x,r) denote the number of sets t._(i) that have non-empty intersection with B(x, r). It follows from the property of the Moran cover that N(x, r) :.::; M, where M is a Moran multiplicity factor. By {13.5) and (13.2) we obtain that for every x and every r > 0, N(z,r)
m;.(B(x,r)):.::;
L
j~l
N(z,r)
m;.(t..Ul):.::;
L
j~l
n(:tj)
D2
II >.,.
8
k~
N(x,r) n(zJ)+l
:'S:Cs
L II
i=l
>..,.•:o::;c5N(x,r)r":o::;C5 Mr",
(136)
k=O
where Cs > 0 is a constant. It follows that the measure m>. satisfies the uniform mas.'! distribution principle (see Section 7) and hence dimn m;. ~ s. We turn to the general case when J-L>. is just an equilibrium measure By definition (13.7) h,_., (uiQ) +.~~log >.iodJ-L;. = 0,
Chapter 5
128
where hp(crjQ) ~£his the measure-theoretic entropy. Let us first assume that tt>. is ergodic. Fix c: > 0. It follows from the Shannon-McMillan-Breiman theorem that for J.£>.-almost every w E Q one can find N 1 (w) > 0 such that for any n ~ Nt(w), I'>.(C, 0 ... (w)) ~ exp( -(h- c:)n), (13.8) where C; 0 ;,. ( w) is the cy Iinder set containing w. If the measure IL>. is ergodic it follows from the Birkhoff ergodic theorem, applied to the function slog >.;0 , that for 11>. -almost every w E Q there exists N 2 (w) such that for any n;::: N 2 (w), (13 9)
Combining (13.7), (13 8), and (13.9) we obtain that for J.'>.-almost every w E Q and n;::: max{N1(w),N2(w)}, n
P.>.(C;0
i,.(w))
~
II >.;
3
n •
exp(2en) ~II >.;i s-o.,
j=O
j=O
where a= 2ef mini log(l/>.J) > 0. This implies that for Jt>.-almost every wE Q and any n;::: max{N1(w),N2(w)}, n
J.L>.(C; 0 ;,(w)) ~II>.,:-"'.
(13.10)
j=O
If J.'>. is not ergodic, then (13.10) is still valid and can be shown by decomposing J.L>. into its ergodic components. Given£> 0, denote by Qt ={wE Q: N 1 (w) ~ l and N 2 (w) ~ £}. It is easy to see that Qe C Qt+l and Q = U~ 1 Qt (mod 0). Thus, there exists lo > 0 such that J.L>.(Qe) > 0 if l;::: io. Let us choose l;::: io Given 0 < r < 1, consider a Moran cover i.I,.,Q 1 of the set Qe. It consists of sets c£0), j = 1, .. . ,Nr,l for which there exist points Wj E Q such that cyJ = C·ZQ 'Zn(w.i) · Set A l.(B(x, r)
n x(Qe)) ::;
J=l
~
N n(xj)
2.: m>..(A~j)) ~ 2.: II >.;. •- J=l k=O
K2N(x, r, l)r•-or ::0 K2Mr"-"'.
Since m>.(x(Qt)) > 0 by the Borel Density Lemma (see Appendix V) for m>..almost every x E x(Qt) there exists a number r 0 = r 0 (x) such that for every 0 < r ~ r 0 we have m>.(B(x,r))::; 2m>.{B(x,r)nx(Qt)).
Dimension of Cantor-like Sets and Symbolic Dynamics
This implies that for any l > £0 and
!lmJx)
= lim logm~(B(x, r)) r-+0
log r
m~-almost
129
every x E x(Qt),
~ lim logm~(B(x, r) n x(Qt)) ~ r->0 log r
8
-'
_a
Since the sets Qt are nested and exhaust the set Q (mod 0) we obtain that !lm.x (x) ~ s~ -a for m~-almost every x E F. This implies that dimn m~ ~ 8~ -a. Since a can be arbitrarily small this proves that dimn F ~ dimn m-' ~ 8~. Note that J.l.\ is an equilibrium measure corresponding to the function s-' log A;0 • Therefore,
•
and the third statement follows.
In view of Statement 2 of Theorem 13.1 the measure m-' is an invariant measure of full Caratheodory dimension (see {54)). We call it simply measure of full dimension. The next statement provides an upper estimate for the number s-'.
Theorem 13.2. Let F be the limit set for a geometnc construction {CPW1CPW4) modeled by a symbolic dynamical system (Q,a). Then (1) [PWl) < hQ(a) 8 ~ - - log Amax ' where Amax =max{.>.~: : 1 ~ k ~ p} and hQ(a) is the topolo!llcal entropy of a on Q; equality occurs if A; = A fori = 1, ... ,p; in particular, if hq(a) = 0, then
(2) [Fu] if>.,=>. fori= 1, ... ,p, then
. F d!IJlH
F = -d. F = 8~ = -hQ(a), . = d' _!!l!B IIDB - 1ogl\
Proof. It follows from Statement 3 of Theorem 13.1 that s~
h,..,(aiQ)
= - J log A; dJ.L.\ 0
hq(u)
< - -log Amax .
The case of equality is obvious. If .>.; = A. for i = 1, ... , p, then J.l-' is a measure of maximal entropy (since the function . is constant). Thus, h~-', (u IQ) = hQ(u). This proves the desired results. •
Chapter 5
130
b
a Figure 2.
SIERPINSKI GASKETS
a) A Simple Construction, b) A Markov Construction. We consider two special cases of symbolic geometric constructions - simple geometric constructions and Markov geometric constructions, specified by the full shift and a subshift of finite type respectively. Examples are shown on Figure 2. This is the well-known SierpiDski gasket - the limit set for a Moran geometric construction (CM1-CM5) on the plane with >. 1 = >.2 = >. 3 < ~ and p = 3. The case of a simple construction is shown on Figure 2a, and thE' case of a Markov
:::::::c~i:nro:::~:: a2:: ;::r:o:~::::;:r::r::::~r~:sisw(hfe for)n~s 1 1 1 We consider the particular case of a subsh!ft of finite type (~1.u). Given p numbers 0 < At. ... , Ap < 1, we define a p x p diagonal matrix Mt(>.) diag(>. 1t, · , >./). Let p(B) denote the spectral radius of the matrix B.
Theorem 13.3. (1) Let F be the limit set for a geometnc con.~truction (CPW1-CPW4) modeled by a subshift of finite type (~1,u). Then dimH P
= dim8 F = dimBF = S).,
=
where S>. tS the unique root of the equation p(AMt(>.)) 1. (2) Let F be the limit set for a ge.ometnc c.onstruction (CPW1-CPW4) modeled by the full shift. Then
dimH F
= dim 8 F = dimBF = 8>.,
where S>. is the unique root of the equation
Proof. The desired result follows from Theorem 13.1 and Theorem A2.8 (see • Appendix II).
Dunension of Cantor-like Sets and Symbolic Dynamics
131
For instance, for the Sierpinski gMkets shown on Figures 2a and 2b with we have respectively that S>.. = log3(log2)- 1 and S>.. =log ¥(log2)- 1 . A more delicate question in dimension theory is whether the Hausdorff measure of the limit set at dimension is finite. In general, the answer is negative. Below we will show that it is positive provided the mea.'lure It>.. is a Gibbs measure (see Condition (A2.20) in Appendix II). If a geometric construction is Markov then the measure It>.. is Gibbs provided the transfer matrix is transitive, i.e., the subshift is topologically mixing (the c!l.'le of an arbitrary transfer matrix can be reduced, in a sense, to the case of transitive transfer matrix; see Appendix II). A more general class of geometric constructions, for which P,>.. is still a Gibbs measure, includes geometric constructions modeled by mixing sofic systems or by mixing subshifts satisfying specification property (see the definition in [KH]). The latter form a much broader class of geometric constructions than the class of geometric constructions modeled by subshifts of finite type.
>. =
!
Theorem 13.4. [PW1] Let F be the limit set for a geometnc construction {CPW1-CPW4) modeled by a symbolic dynamical system (Q,u). Assume that the measure m>.. is a Gibbs measure then (1) the measure /1->.. satisfies the uniform mass distnbution pnnciple; (2) the Hausdorff measure mH(·, s,.) is equivalent to the measure m,.; in particular, 0 < mH(F,s>..) < oo; (3) 4m,..(x) = dm~(x) = S>. for every x E F; (4) dimH(F n U) = S>. for any open set U which has non-empty intersection with F.
Remark. If p, is an arbitrary Gibbs measure on Q, then by Theorem 15.4 below its push forward me!l.'lure m = X•/1- is exact dimensional (see definition of exact dimensional me!l.'lures in Section 7) and the pointwise dimension of m is constant almost everywhere (and is equal to h~-'(uiQ)/ fqlog>.i 0 dJ.t).
Proof of the theorem. Since It>. is a Gibbs measure one can repeat arguments
ill the proof of Theorem 13.1 (see (13.5) and (13.6)) and conclude that it satisfies the uniform mass distribution principle. This proves the first statement. Moreover, (13.6) implies that S>. ~ 4m~ (x) for every x E F. It follows that B>.. ~ dimH F and mH(F, S>.) > 0. We now prove that m,.(·) ~ const x mH(·,s>.)· Given 6 > 0 and a Borel subset Z c F, there exists e > 0 and a cover of Z by balls Bk of radius rk ~ e satisfying k
It follows from (13.6) that
m,..(Z) ~
E m,.(Bk) ~ G1ME (rk)'~ ~ G1MmH(F, B>.) + G1M6, k
k
where G1 > 0 is a constant. Since tJ is chosen arbitrarily this implies that m>.(Z) ~ G1MmH(F, S>.)
Chapter 5
132
We now show that mH(·, 8>.) :::; const x m~() Let Z c F be a. closed subset. Given 5 > 0, there exists c > 0 such that for any cover U of Z by open sets whose diameter :::; e we have
mH(Z,8~):::; L(diamU)"~ +5.
(13.11)
ueu
Note that one can choose a cover U of Z by basic sets b. {k) diam.b,.(k) :::; c and m>.(b.(k)):::; m>.(Z) + .t.eu
L
= 6;0
;,. k 1 1
satisfying
o.
We can apply (13.11) to this cover U and obtain using (13.1) and (13.5) that
L
mH(Z, 8>.):::;
(diamt.{kl)•>
.t,.!klEU
:::; K2D} 1
L
n
n(k)
L
+ 8:::; K2
Ai; ·~
+5
.t.C•>eu j=O
m.>,(.b.(k))
+ o 5 K2D1 1 m>.(Z) + (K2Di 1 + 1)!5.
.t.eu
Since 5 is chosen arbitrarily this implies the second statement. Fix 0 < r < 1. For each wE Q choose n(w) according to (13.2). It follows from (13.1) that b.;. in(w)+l c B(x, K2r), where X= x(w). By virtue of (13.5) for all wE Q, n(w)+l
m>.(B(x,K2r)) ~ m>.(.b.; 0
;.
1.>+,)
~ D1
fl
>..::
~ C2r"',
k=O
where c2 > 0 is a constant It follows that for all -d
m, (X) =
X E
F,
. log m,x(B(x, r)) 11m l 5 S,x.
r--+0
og r
This implies the third statement. We now prove the last statement. It follows from the second statement that mH(F n .b.;. in' 8>.) > o. Thus, dimH(F n b..., in) ~ S,x. Now, let u be any open set with F n U =F 0. If x = x(i0 i 1 ... ) E F n U and n > 0 is sufficiently large then b.; 0 in C U. Therefore,
This completes the proof of the theorem.
•
Dimension of Cantor-like Sets and Symbolic Dynamics
133
Self-similar Constructions There is a special class of geometric constructions of type (CG1-CG2) which are most studied in the literature (see for example, (Fl]) -self-similar geometric constructions. They are geometric constructions with basic sets Aio . ;. given as follows: where h 1 , •.• , hp: D -t D are conformal affine maps (i.e., maps that satisfy dist(h;(x),h.(y)) = >.;dist(x,y) for any x,y ED, where Dis the unit ball in lim). Here 0 < ..\; < 1 are ratio coefficients. These geometric constructions can be modeled by an arbitrary symbolic dynamical system (Q, o'). Clearly, self-similar constructions are a particular case of Moran geometric constructions with stationary ratio coefficients (CPW1-CPW 4). Therefore, by Theorem 13.1, the Hausdorff dimension and lower and upper box dimensions of the limit set of a self-similar construction coincide. The common value is the unique root of Bowen's equation PQ(s ••. , 'Yp), 0 < 'Yi < 1, i = 1, ... ,p, consider a.
134
Chapter 5
Moran cover i4 = ilr('Y) = {~(j)} of the limit set F constructed in Section 13. Given an open Euclidean ball B(x, r) of radius r centered at x, denote by R(x, r) the number of sets ~(j) that have non-empty intersection with B(x,r). We call a vector 1 estimating if R(x, r) ::; constant (14.1) uniformly in x and r. We call a symbolic geometric construction (CGI-CG2) regular if it admits an estimating vector If 'Y = ('Yt, ... , 'Yp) is an estimating vector for a regular geometric construction, then any vector 1' = (1'1> ... , "Yp) for which')'; 2:: .:y., i = 1, ... ,pis also estimating. We provide an example of a regular geometric construction on the plane that illustrates how the choice of the estimating vector can be made.
A.
'
::
~ 1
I
:
I
I
------vr---: :_:
' t-'
I I
::
\ I
I
I I
I
~ ------
o~----~+-r4-+.---------1Hr;~~r----;
-A.
---- ------- _'D~-. I 1
I 1
I 1
1 I
I
I I
I I
I I
I I
1 1
I
I I
I I
I I
I
I
I
I
I I
-I~----~---~---~-1----·L-----~---~---~--·--1.&------~ Figure 3. A REGULAR GEOMETRIC CoNSTRUCTION. Example 14.1 (PWI) Let 'Yt.'Y2,'Y3, and ,\ be any numbers in (0, 1). Given a number i = 1,2,3 consider a simple geometric construction (CG1-CG2) on the interval [0, 1) x {i- 2} with 2n basic sets of size 'Yin at step n. We denote this construction by CG(-y;). Since the 2n intervals at step n in each of these constructions are clearly ordered we may refer to the ith subinterval at step n, 1 ::; i ::; 2n of these constructions. Consider the 2n polygons in (0, 1] x [-1, 1) having six vertices which consist of the two endpoints of the ith subinterval at step n for all three constructions. We define the 2n basic sets at step n by intersecting these 2" polygons with the rectangle [0, 1] x (-,\", ,\n). This produces a simple geometric construction (CG1-CG2) on the plane See Figure 3.
Dimension of Cantor-hke Sets and Symbolic Dynamics
135
It is easy to sec that the limit set F of this geometric construction coincides with the limit set of the construction CG(-y2 ). Hence, by Theorem 13.3, dimH F = _ 1;:,~~2 and does not depend on 'Yt. 1 3 , or A. Let us choose numbers 'Yt, 12, 'Y3, and A such that 'Y2 < 'Yt = 'Y3 < A and 'Y2 < kn or 'Y2 < >.13 One can see that the inscribed and circumscribed balls of the basic sets at step n have radii wluch are bounded from below and above by cnr and c2>.n, respectively, where Ct and C2 are positive constants which are independent of n. Thus, these balls cannot be used to determine the Hausdorff dimension of the limit set. • Consider the positive number s-y such that Pq(s-ylog'Y; 0 ) = 0, where Pq denotes the topological pressure with respect to the shift u on Q (see Section 11) Let P.-r denote an equilibrium measure for the function (ioit ... ) >-+ s"''log'Y.., on Q, and let m"'' be the push forward measure ou F under the coding map x (i.e., m"''(Z) = p."''(x- 1 (Z)) for any Borel set Z c F). The following result provides a lower bound for the Hausdorff dimension of the limit set. Its proof is quite similar to the proof of Statement 2 of Theorem 13.1
Theorem 14.1. [PW1] Let F be the l1mit set for a regular symbolic geometnc construction. Then dimH F :::: s"'' for any estimating vector 1 Hence, dimH F :::: sup 8"'1, where the supremum is taken over all estimating vectors 'Y. In the case when the measure J.t. 1 is Gibbs one can strengthen Theorem 14.1 and prove a statement that is similar to Theorem 13.4.
Theorem 14.2. [PWl] Let F be the limit set of a regular symbolic geometnc construction and 'Y an estimating vector. Assume that the measure fJ."'( is a Gibbs measure Then
(1) the measure m"'' satisfies the uniform mass distnbution principle; (2) 0 < mH(F,s"''); moreover, m"''(Z) :0:::: CmH(Z,s"'') for any measurable set Z C F, where C > 0 is a constant; (3) s"'' :0:::: 4m~(x) for every x E F; (4) dimH{FnU):::: s"'' > 0 for any open set U which has non-empty intersection with F.
The second statement of Theorem 14.2 is non-trivial only when s 1 = dimH F. Otherwise, mH(F, s-y) = oo. If 8-y < 8 = dimH F, then the s-Hausdorff measure may be zero or infinite. Theorem 14.2 holds for simple geometric constructions or Markov geometric constructions with transitive transfer matrix In [Barl], Barreira gave sufficient conditions for a geometric construction to be regular. Roughly speaking, it requires that the basic sets rontatn sufficiently large open balls. We begin with geometric constructions on the line.
Theorem 14.3. Assume that each basic set 6.; 0
in of a symbolac geometnc construction {CG1-CG2} on the line contains an interval 1;0 in of length 0 < .>."" in < 1 such that 1;0 in n Iio in = 0 for any (io .. in) # (jo ... in). Assume also that there extSts 0 < 'Y < 1 such that
136
Chapter 5
where the minimum is taken over all Q-admissible n-tuples (i 0 .•. in)· Then the geometnc construction is regular with the estimating vector (re-•, ... , -ye-•) for any e > 0.
Proof. Given e > 0, we have A.., ,n > (~re-")n for every (i 0 i 1 .. ) E Q and any sufficiently large n. Given r > 0, one can find a unique number n = n(r) > 0 such that For any interval I of length r there exist at most two basic sets of length 2: ('Ye-•)'• intersecting I. Therefore, for every point x in the limit set the number R(x, r) in the definition of regular geometric constructions (see (14.1)) we obtain that R(x,r)::; 2 for all sufficiently small r. Hence, the construction is regular with the estimating vector (-ye-•, .. . , -ye-•). • We now formulate a criterion of regularity for a geometric construction in Rm with m > 1 Fix a point x E F and a number A. > 0. Given n > 0, consider two basic sets 6., 0 •n and 6.;0 in intersecting the ball B(x, A.n). Denote by a(6.; 0 in, fl; 0 in) the minimum angle of spherical sectors centered at x which contain both 6.; 0 in and 6.,0 Jn. Let an (x, A.) be the minimum of all the angles a( 6.;0 in, 6-;o ;,J.
Theorem 14.4. Assume that each basic set of a symbolic geometnc comtntction (CG1-CG2} in IR"' with m > 1 contains a ball B; 0 in C ilia in of radius An wzth 0 0 such that an(x, A.) 2: liN' for all x E F and n 2: 1 and that B; 0 in nB, 0 Jn = 0 for any (io ... in)# (jo . . jn)· Then the geometnc construction is r·egular with the estimating vector (A., .•. , A.).
Proof. By elementary geometry there exists a universal constant C = C(li) such that the maximum number of sets 6.; 0 ;n intersecting a ball B(A.n) does not exceed C. This proves the result. • ConsidPr a simple regular geometric construction inlR"" with the limit set F. It follows from Theorem 14.2 that the Hausdorff dimension of any open set U which has non-empty intersection with F satisfies dimH(F n U) 2: s for some .~ > 0. The following example shows that the converse statement may not be true.
Example 14.2. [Earl] For each s E (0, 1), there exists a geometric construction (CG1-CG2} on the line modeled by the full shift (~t, a) such that (1) dimH(F n U)
= s for any open set U with F n U #
0;
(2) the construction is non-regular. Proof. Define the function m· Q
m (w) =
. .
m (totl-··
-t N U { +oo} by
) = { +oo,
w=O
least j E N with ii = 1, w
#
0.
137
Dimension of Cantor-like Sets and Symbolic Dynamics 0
+t
1---------1110
\
!.. 1(4)
~
A
_ A(2)
1----t Ll 01 -
Figure 4. A
NON-REGULAR GEOMETRIC CONSTRUCTION.
Define also the numbt>rs A; 0
Tij= 0 A;
A;o in =
Ll
;,.
3
by
n 0. We consider basic sets spaced as shown on Figure 4. They have the following property: if D.>o. ;. =[an, bn] then an E D.io ;.o and bn E .1.;0 • in 1· Define intervals ,1.0) = .1.;0 ; 1 , where (io .. i;) = (0 .. 01). Inside each D,!i) we have a sub-construction modeled by CEt, a) with rates Ao Olio ;nf Ao 01 = TI~= 0 A;,(j + 2). Therefore, the Hausdorff dimension ofF n D,(i) is equal to s, where s is the unique root of equation (14.2), with j + 2 instead of j (sec Theorem 13.3). Hence, dimn(F n D,Ul) = s. Since F = {0} u U;>o(F n D,CJ)) it follows that dimH F = supi>O dimH{F n AUl) = s. Now, ifF n U # 0, there exists x = x(i 0 i 1 •.. ) E (F n U) \ {0}, and n > 0 such that A; 0 ;,. cU. Hence, .~ = dimn F ;::: dimn(F n U) ;::: dimn(F n A 00 .;,. ) = s This proves the first statement. Consider a vector b1.'Y2). For each n > 0, set rn = 2/{n -1) 1• Select now the Smallest positive integer k that > Tn and n;:~ 'YiJ ::; Tn for 1 some (i 0 i 1 . ) E Et Set 'Y = minb1.'Y2} and observe that 'Yk+l::; rn Hence, k ;::: logrn/ log')'-1 Therefore, for kn = log rn/ log')'-1 we get R(O, rn) 2: 2kn-n {where R(O,rn) il:i defined in {14.1)). We also have
SUCh
k n
-n> log2-log(n-l)l _ 1 _n log 'Y
n;=O ')';
~ n--+oo
logn!
---n -log')'
~
n--+oo
logn! -log')'
Chapter 5
138 Therefore, there exists D > 0 such that kn - n obtain R(O,rn)
~
2Dlognl
~
= n!Dlog2 >
D log n! for all n > 0, and we
(
r:
Dlog2 )
As the sequence 1·n decreases monotonically to 0 and (-n, 12) is arbitrary we proved that the construction is non-regular. • If a geometric construction is regular one can effectively replace its basic sets by balls to obtain a lower bound for the Hausdorff dimension of its limit set (see Theorem 14.1). We further exploit this approach and show that, under some mild assumption, a geometric construction (CG1-CG2) (whose basic sets are, in general, arbitrary close subsets and are possibly intersecting) can be effectively compared with a geometric construction whose basic sets are disjoint balls.
Theorem 14.5. Let F be the limit set of a qeometric construction (CGJ-CG2} modeled by a symbolic dynamical system (Q, u). Assume that there e:nst numbers A1 , ...• Ap, 0 < Ai < 1 such that for any admissible n-tuple (i 0 ..• in), ii = 1, ... ,p, we have n
diamD.to
in
:$ C
IJ Aii,
j=O
(14.3)
where C > 0 is a constant. Then (1) there e:nsts a self-similar geometnc construction modeled by a symbolic dynamical system (Q, a) satisfying: a) its basic sets B; 0 '" are disjoint balls, b) diamB; 0 in = 2C f17=o A;,, and c) there i.~ a Lipschitz continuous map 'lj; from its limit set F onto F; (2) dimBF :$ S:>...
Proof. Consider a self-similar geometric construction with ratio coefficients A1 , ... , Ap modeled by a symbohc dynamical system (Q, a) whose basic sets B'o in are disjoint balls of radii A;j. Denote its basic set by F. Let x: Q -+ F and Q -+ F be the coding maps. Consider the map 'lj; = xox- 1 : F-+ F We shall show that 'lj; is a (locally) Lipschitz continuoub map. Choose x, y E F 1{x) = (ioi1 .. ), 1 with p(x, y) e: We have that (y) = (joj1 ... ), and io = jo,. 'in = jn, in+l # jn+l for some n > 0. Therefore, llx- Yli ~ cl ITJ'=o A;j, where C1 > 0 is a constant. One can also see that 'lj;(x),'lj;(y) E .6.;0 in· Hence, A;r Since the map 'lj; is onto this proves the by (14 3}, li¢(x)- 'lj;(y) II c first statement. The second ~tatement follows immedi.ttely from the first one. •
c n;=O
x:
s
x-
x-
s n;=O
Geometric Constructions with Ellipsis We describe a special class of regular geometric constructions We say that a geometric construction (CG 1-CG3) in 1R2 is a construction with ellipsis (see Figure 5) if each basic set A; 0 in is an ellipse with axes }t/2 and X'/2, for some 0 ..), where A is any number in the interval {O,t,.2(J.). Proof. For each >.. E (0, I) define the function 9>.= (0, 1) x (0, 1) -+ R by
[(X/t,.2)tt,.]tf2, >..<X Y>.(t,.,X)= (X/t,.)t, >..=X { t,.t-t, >.. >X where t =:log },_flog A. Consider an ellipse En with axes };n /2 and X"n /2, where a= log },_flog X. We assume that it is located outside the ball B(O,t,.n), is tangent to this ball at a. point, and that the major axis of the ellipse points towards 0. Denote by 2/3{n) the smallest angular sector centered at 0 that contains En· The desired result follows immediately from Theorem 14.4 and the following lemma.
Lemma. For each A E (0, X), there exists a number C > 0 such that tanf3(n) ~ Cg>.(t,., X)-n as n -+ oo. In particular, f3(n) decreases exponentially with the mte Y>.(t,.,X)- 1 when },.2/X < .>.<X and is uniformly bounded away from 0 if
o < .>. s. !//X.
Figure 5. A CONSTRUCTION WITH ELLIPSIS. Proof of the lemma. Consider an orthogonal coordinate system centered at 0 with the x-axis directed along the major axis of En. If m(n) = tanf3(n) is the slope of a line starting at 0 and tangent to En, the points of tangency are solutions of the equation
140
Chapter 5
provided that the discriminant of this equation is zero:
Set b01 =}:,_f)(>. One can see that 4 m(n)2 following cases:
= b~"'n/[b~(b~ + 1)].
We .(6., Xf". It is easy to check that if ~2/ X< >.<X, we have 9A(6., :\) > 1 and ifO < >.::;~//X, we have 9>.(6., A)::; 1. Since tanx rv x as x ~ 0 the desired result follows. • (a) (b)
15. Moran-like Geometric Constructions with Non-stationary Ratio Coefficients In this section we study Moran-like geometric constructions with ratio coefficients at step n depending on all the previous steps. This class of geometric constructions was introduced by Barreira in [Bar2]. Consider a geometric construction modeled by a symbolic dynamical system (Q, u) We assume that uiQ is topologically mixing and the following conditions hold: (CBl} ~io i,.j C ~io ;,. for j = 1, ... ,p; (CB2) where !1.; 0 ;ft and Bio ;~ are closed balls with radii K1r; 0 ;,. and K2r; 0 ,;,. and 0 < K1 S K2; (CB3) intl1.io i,. n intl1.jo im = 0 for any (io ... in) =F Uo ... jn) and m 2: n. The class of geometric constructions (CB1-CB3) is quite broad and includes geometric constructions (CPW1-CPW4) (see Section 13}, geometric constructions with contraction maps (see below}, and more general geometric constructions with quasi-conformal expanding induced maps (see Theorem 15.5 below). The study of geometric constructions (CB1-CB3} is based upon the non-additive version of the thermodynamic formalism (see Section 12 and Appendix II). We dt>fine the sequence of functions cp = { 0 the number of basic sets .t..io ;• = x(C; 0 '•) in a Moran cover U,. that have non-empty intersection with the ball B(x, r) is bounded from above by a number M, which is independent of x and r (a Moran multiplicity factor). Repeating arguments in the proof of Statement 1 of Theorem 13 1 one can prove the following result Theorem 15.1. [Bar2] Let F be the limit set of a geometnc construction (CB1C!J3) modeled by a symbolic dynam~cal system (Q, cr). Assume that the numbers 7'; 0 .;. satisfy Condition {15.8}. Then §. ~
dimH F :::; dim8 F ::=; dimBF
~
s.
The second assumption we need is the following: the sequence cp is sub-additive, i.e., nkcpo This implies that
.n)) = log2 + n (>o in) C;o '• n n
where C'O ;. are cylinder sets. By definition CPE+(srp) = log2 +slim An • n-too n
A.nh
= -n~c(a, b).
Therefore,
?: log2- slim n~c(a, b) ?: log2- sa k->oo
nJ..
Moreover, we have that -bn ~An ~ -an and thus, CPE+(srp) == log2- sa. 2 This implies that 8 = log2/a. •
Pointwise Dimension of Measures on Limit Sets of Moran-like Geometric Constructions Let v be a Borel probability measure on the limit set F of a geometric construction (CB1-CB3} modeled by a symbolic dynamical system (Q,u). We formulate a criterion that allows one to estimate the lower and upper pointwise dimensions of v. Given x E F, set
. f . log v{A; 0 .. i.) _ x ) =m , d( 11m n->oo logjA; 0 .i.l -( X ) =Ill . f -liD . log v(A;0 .i.) d 1 , n-too log IA'O in I where IA;0 ; . I denotes the diameter of the basic set A; 0 taken over all w == (ioit ... ) E Q such that x = x(w).
;.
and the mfimum is
Theorem 15.3. Assume that a geometric construction (CB1-CB3} satisfies Conditzon (15.3). Then (1) tfv(x) ~ d(x) for all x E F; (2) .d_(x) ~ fl.,(x) for v-almost all x E F; -
dcl
-
= d(x) == d(x) for v-almost every x E F, then !l,(x) == d,(x) == d(x) for v-almost every x E F.
(3) if 4(x)
Chapter 5
144
= x. Given r > 0, choose n = n(r,w) such that IA..,.;nl oFa,c coincides with the set of points for which d.(x) ? a. We will show that 4.,(x) ? a for almost every x E Fa· This will imply Statement 2. Indeed, if fl(x) > !l.,(x) on a set of positive measure then there exists a such that !l(x) >a> 4.,(x) on a set of positive measure. Fix x E Fa,c and r > 0. Consider a Moran cover 11,. of the set Fa,C and choose those basic sets in the cover that have non-empty intersection with the ball B(x, r) By the property of the Moran cover there are points xi E Fa,c, j = 1, ... , M (where M is a Moran multiplicity factor which is independent of x and r) and basic sets A,(;) such that Xj E .6,0) = A; 0 ·'·' diamA,(i) ::; r, and M
B(x,r)nFa,C
c
U 0. By the Borel Density Lemma (see Appendix V) for v-almost Pvery x E Fa,C there exists a number ro = ro(x) such that for every 0 < r ::; r 0 we have v(B(x, r)
n F)::; 2v(B(x, r) n Fa,c)
This implies that for v-almost every x E Fa,c,
. logv(B(x,r)) > . logi:;: 1 v(A.0
log r
logi:;: 1 C[A;0 '•Ia > '·mlogi:;: 1 Ctr"' ll =a, log r - r->0 logr where Ct > 0 is a constant. The last statement is a direct consequence of the preceding statements. .
> hm -
r->0
•
Dimension of Cantor-like Sets and Symbolic Dynamics
145
Let v be a Borel probability measure on the limit set F of a geometric construction (CB1-CB3). Even if the pointwise dimension dv(x) of v exists almost everywhere it may not be constant and may essentially depend on x. AB Example 25.2 shows, the pointwise dimension may not be constant even if v is a Gibbs measure. This phenomenon is caused by the non-stationarity of geometric constructions (CB1-cB3). The situation is different for geometric constructions (CPW1-CPW4) where ratio coefficients do not depend on the step of the construction. Theorem 15.4. Let F be the limit set of a geometnc construction {CPW1CPW4) modeled by a symbolic dynamical system (Q, a) and fl. an ergodic measure on Q. Let also m be the push forward measure of p, to F. Then m is exact dimensional {see Section 7} and for p,-almost every w = (ioit ... ) E Q we have 4m(x) where
X=
-
h,.(qJQ)
= dm(x) = _ J.Q Iog A·•o d1-' ,
X(W).
Proof. Since p. is ergodic by the Birkhoff ergodic theorem applied to the function w >-+ logA"' (where w = (ioit .. ) E Q) we have that for (l-almost every w the following linut exists: lim -1
L: log A,. =1log A; n
0
dJ.L.
Q
n->oon k=O
Exploiting again the fact that p. is an ergodic measure, by the ShannonMcMillan-Breiman theorem we obtain that for J.L-almost every w = (i 0 i 1 .•. ) E Q,
It follows from Condition (CPW3) that for J.L-almost every w = (ioi 1 ... ) E Q, lim
.!. log diam A;
n->oo n
0
= lim
i n
n->oo
t
.!.n k=O log A;. =
llog A; 0 dp, Q
The desired result follows now from Theorem 15.3.
•
Geometric Constructions with Quasi-conformal Induced Map Let F be the limit set of a geometric construction (CG1-CG3) in Rm modeled by a subshift (Q, 0'). Since we require the separation condition (CG3) the coding map x: Q -+ F is a homeomorphism and the induced map G: F -+ F is well defined by G = X o 0' o x- 1 We have the following commutative diagram
Chapter 5
146
It is easy to see that G is a continuous endomorphism onto F. By the result of Parry [Pa] it is a local homeomorphism if and only if the subshift is a subshift of finite type, i.e. Q = ~1, where A is a transfer matrix. From now on we consider tllis case and assume that A is transitive, i.e., the shift is topologically mixing (see Appendix II). The induced map encodes information about the sizes, shapes, and placement of the basic sets of the geometric construction and hence can be used to control the geometry of the construction. In order to illustrate this let us fix a number k > 0. For each w = {ioil· . ) E ~1 and n 2:: 0, we define numbers A(w n)
- '
= A (w ~
'
"X(w n) =X (w n) =sup { ,
k
an(y)ll} llx- Yll ' IIGn(x)- G"(y)ll} llx- !Ill ,
n) = inf { IIG"(x) -
,
(15.7)
where the infimum and the supremum are taken over all distinct x, y E F n ~; 0 in+•. It may happen that ~(w, n) = 0 or "X(w, n) = oo for all sufficiently large n. In the case 0 < .d(w,n) ~ X(w,n) < oo, consider the limits
.d(x)
1 = nlim -log~(w,n), ...... oo n
-
A(x)
1 = n-+!Xl lim -logA{w,n), n
where x = x(w). Notice that by the multiplicative ergodic theorem (see for example [KH]), the limits exist for almost all x with respect to any Borel Ginvariant measure p, on F provided that
If the map G is smooth the numbers ~(x) and X(x) coincide with the largest and smallest Lyapunov exponents of Gat x (see definition of Lyapunov exponents in Section 26). When G is continuous these numbers can serve as a substitution for the Lyapunov exponents (see [Ki]). We consider the case when the trajectories of the induced map are strongly unstable. More precisely, we call the induced map expanding if there exist constants b 2:: a > 1 and ro > 0, such that for each x E F and 0 < r < ro we have B(G(x),ar) C G(B(x,r)) C B(G(x),br). (15.9) Note that if the induced map G is expanding then it is (locally) hi-Lipschitz. Furthermore, if the induced map G is expanding then the placement of basic sets of the geometric construction cannot be arbitrary (sec Theorem 15.5 below). We now specify the choice of the number k in (15 7). Namely, we assume that k is so large that (15.10) diam ~io in+• ~ ro.
Dimension of Cantor-like Sets and Symbolic Dynamics
147
We say that an expanding induced map G is quasi-conformal if there exist numbers C > 0 and k > 0 (satisfying (15.10)) such that for each w E E1 and n ~ 0, X(w, n) :s; C ~(w, n). (15.11) As the following example shows geometry of constructions (CG1-CG3) with expanding quasi-conformal induced maps (i.e , the placement of basic sets and their "sizes") is sufficiently "rigid".
Theorem 15.5. Let F be the limit set of a geometnc construction (CG1-CG3} in IRm modeled by a subshift of finite type (E!, u). Assume that the induced map G is quasi-conformal and that the basic sets on the first step of the construction have non-empty interwrs. Then (1) the construction satisfies Conditions (CB1-CB3}, i.e., it is a Moran-like geometnc construction with non-stationary ratio coefficients; moreover, it also satisfies Conditions (15.3} and (15.5}; (2) dimy F = dimBF == dimBF = s, where s ib a unique number satisfying lim !.tog n-+oon
(diam(Fn~; 0
""' L..J
(io
;.))'
=0.
i.)
E1-admissible
Proof. We outline the proof of the theorem. Smce the basic sets on the first step of the construction have non-empty interiors we observe that each basic set ~io ;,. of the geometric construction satisfies
!J..io
in C
~io
.in C B;o
in'
where !1.;0 ; . and B; 0 ; . are closed balls with radii [.0 there exist numbers Ct > 0 and C2 > 0 such that
c1 - - < rX(w, n) - -•o
;.
and 1';0
; ••
Moreover,
< r·•o '•· < - c2 -- ~(w, n) ·
'• -
Since the induced map G is quasi-conformal this implies that the geometric construction satisfies Condition (CB2). Clearly, Conditions (CB1) and (CB3) hold and thus, the construction is a Moran-like geometric construction with nonstationary ratio coefficients. By straightforward calculations one can show that given w = (i 0 i 1 ... ) E E1 and n,m ~ 1, ~(w, n + m) ~ ~(w, n) x ~(un(w), m), and similarly, X(w, n
+ m) s
X(w, n)
X
X(an(w), m).
Since the induced map G is quasi-conformal the above inequalities imply Condition (15.5). It follows from (15 9) and (15 10) that a :s; ~(w, 1)
:s; X(w, 1) :s; b.
Thus, the first inequality in (15.3) holds. Similar arguments show that the second inequality also holds. This implies the first statement The second statement follows from the first one and Theorem 15.2. •
148
Chapter 5
One can build a geometric construction (CPW1-CPW4) for which the induced map on the limit set is not expanding: whether it is expanding depend on the placement of basic sets on each step of the construction (see below). We present now more sophisticated examples which illustrate properties of induced maps. Example 15.2. [Bar2] There exists a geometnc construction {CG1-CG9} on the line modeled by the full shift (Et, u) such that:
(1) each basic set ~;0 ; , is a closed interval; (2) there e:nsts a point w E Ef such that ~(w, n)
= 0 and A(w, n) = oo for each n;::: 1; hence, the induced map G is not expanding.
Proof. Consider a geometric construction on the line modeled by the full shift on 3 symbols for which ~;0 ; , = [ajn• bjn] for each (io ... in) = (0 ... 0} and j = 0, 1, 2. One can choose the basic sets such that: (a) for each n ;::: 0 the points bon, a 1n, b1n, and b2n lie in the limit set F; (b) the difference a 1, - bon is e-a(n+l) for n even and e-b(n+l) for n odd; (c) the difference a2,.. - btn is e-a(n+l) for [n/2] even and e-b(n+l) for [n/2] odd. Here a and b are positive distinct constants (see Figure 6a where a = log 5, b = log6, and the intervals ~; 0 ;,. are of length 5-n). This implies that ~(w, n) = 0 and A(w, n) = oo for the point (ioi 1 .•• ) = (00 . . ) and all n;::: 1. • The following example shows that there are geometric constructions (CG 1CG3} for which the induced map may be expanding but not quasi-conformal. It also illustrates that in this case Theorem 15.5 may fail. Example 15.3. [Bar2) There extsts a geometnc constructwn (CGI-CG3} on the line modeled by the full shift (Et, u) such that
(1) each basic set ~; 0 ; , is a closed interval of length depending only on n; (2) the induced map G is expanding but not quasi-conformal; (3) dimy F = dim 8 F < dimnF.
Proof. (See Figure 6b.) Let An be numbers defined by (15.6). Consider a geol!letric construction (CG1-..,._,.
1+2E~ 1 co.J
A(w, n):::; sup->.- x 1 2 z:::oo -aJ :::; m;:::o e "' i=l e
A(w n)
-
'
e>.,._,. - e>.m
> inf - - x -
m>O
ch"(l+c"') 1- 3e-a
< oo,
1- 2 E~ 1 e-aj e-a"(I- 3e-a) J. > >0 1 + 2 L..,J=l ~ e-aJ 1 + e-a ' 00
for all b ~ a > log 3. If a is sufficiently large we have
It follows that a :::; ~(w, 1) :::; X(w, 1) :::; j3, where a and j3 are some positive constants. Therefore, for a sufficiently small r 0 and any x, y E F we have that allx- vii :::; IIG(x)- G(y)ll :::; J3llx- Yll provided llx- vii :::; ro. Since G is a local homeomorphism one can easily derive from here that G is expanding. Notice that by the construction of the numbers An, we have that sup (Am - Am-n) m;::o for any n
~
= -an,
inf (Am - Am-n) m;::o
= -bn
(15.12)
0. Hence,
This proves that the induced map G is not quasi-conformal. One can see that 2n-l :::; N(F, e>.,.) :::; 2" (recall that N(F, r) is the smallest number of balls of radius r needed to cover F). Therefore, (15.12} implies that .
d --.!illB
F- log2 - -b-'
-d. F 1mB
log2 =--. a
Notice that our construction is a geometric construction (CB1-CB3) with basic sets satisfying (15.3). By Theorem 15.1 we have that dimn F ~ s, where 8 is the uruque root of the equation PE+(stp) = 0. Since A,.~ -bn for all n ~ 0 we • conclude that 8 ~ log 2/b. Hence, dimn F = dim 8 F = log 2/b. •
Chapter 5
150 0
(
J
ao
a!
(J\
\ a2 1 5
5
ao2
aoo ao1
H H 1--t
1
1
36
25
a 0
+(
\+
)! ,an
a 00 .............. +( }
I
_1 { 4
4
} + +{ b Figure 6. GEOMETRIC CONSTRUCTIONS WITH INDUCED MAPS: a) Non-expanding, b) Non-conformal. Geometric Constructions with Contraction Maps There is a special class of geometric constructions (CGl-cG2) that are wellknown in the literature (see for example, (FI]) - constructions with contraction maps. Their basic sets .a.,0 • ;,. are given as follows
.a.;0 . in = h., 0 h1
1 0 • • · 0
h;, (D).
Here D is the unit ball in Rm and hi. . .. , hp: D --+ Dare hi-Lipschitz contraction maps, i.e., for any x, y E D,
A dist(x, y)
~ dist(h;(x), h;(y)) ::;
X; dist(x, y),
where 0 < 41 ::; X;< 1. They arc modeled by a subshift (Q, rr). It is easy to see that these geometric constructions are Moran-like geometric constructions with non-stationary ratio coefficients of type (CB1-CB3) and hence can be treated accordingly.
Dimension of Cantor-hke Sets and Symbolic Dynanncs
151
If we require the separation condition (CG3) then the coding map is a homeomorphism and we can consider the induced map G on the limit set F of the construction which acts as follows G(x) = h,(x)- 1 for each x E F n ~. and i = 1, .. , p. Hence, G is expanding, i.e., it satisfies (15.9). As we mentioned above G is a local homeomorphism if and only if Q is a topological Markov chain, i.e., Q = E1 for some transfer matrix A (which we assume to be transitive). One -7 az+±db cz be = 1 or -1. A linear fractional transformation g is said to be hyperbolic if tr 2g = (a+ d) 2 > 4 and loxodromic if tr 2 g E C\ [0, 4] A (classical) Schottky group r is a Kleinian group with finitely many generators 911 ..• gp, p ~ I which act in the following way: there exist 2p disjoint circles 'Yl, 'Yi, . . , 'Yp, 'Y~ bounding a 2p-connectcd region D for which 9; (D) n D = 0 and gi('Yi) = 'Yj for j = 1, ... ,p. The group r is known to be free and purely loxodromic, i.e., all non-trivial elements of r have either hyperbolic or Joxodromic type (see [Mas], [Kr]).
C h a p te r
5
152
IN G T O SPOND CORRE . N IO T ATORS UC GENER ONSTR T R IC C THREE E H M IT O E W A G GROUP e E C T IO N E if th A REFL
s ta b il iz e r C z t in z such o at a p rhood U nuously ~eighbo ti n a (r) of o s c Q a is h t d e id to a c t r is finite, a n d z for g E r z· T h e s a s is r p ontinuity T h e grou (z) :: z} of z in z a n d g(U•.) "" U. n o f disc io g re g e factor r : j th r. = {g .)E nrU . "" 0 for all g Edisr continuously is aclallSecdhottky group th e n th e ts ic th a t g(U et is a class hich r a c e li m it s th a t if r E C at w wed a s th eral, a re p o in ts z p r I t is known ie . v p e s u b n n e u ca fg gen o f th e g ro closed surface o k y g ro u p onformal b u t, in d) a c al S c h o tt (one-side is ic a re f ss a / y la h b c f) ic ( d a h n le w r e ed s fo d p ll o t a a e c m s m T h e li m it c o n s tr u c ti o n w it h is c o n s tr u c ti o n is S c h o tt k y g ro u p is t th e h a c j T th -r tn . e :: m ng ow "YJ :::: for a g e o ine nor c o n tr a c ti ne c a n sh up, ial case ff "" 3). O h a reflection g ro r th e spec p In re . e e h n e it h e r a p e w it ty th w 7 e o d it e re n te il u fi of a l wh ssocia e Fig subshift r o u p (se ic c o n s tr u c ti o n a long main diagon it s e t o f th e g n o ti c etr m sa th e r e fl e th e geom fe r m a tr ix h a s O m a p G on th e li s n e ra ti n g maps, g e ti n g a n d th e tr a n th a t th e induced xpanding. e ac a re c o n tr equal to L N o te th conformal a n d s ic S e t s o o re a m jo in t B a s is is D n h o entries it ti ) w c o n s tr u c 13) m o d 1-CPW4 geometric e Section e s (CPW (s n ) o f th e 4 ti c W u sic s e ts o W 1 -C P ic C o n s tr ti o n (C P a t th e ba .. • ) E Ejj:, c th u tr e s G e o m e tr m n u o i1 etric c We ass h w = (i 0 er a geom (~1, a). t for eac a We consid ift o f finite ty p e th fy ri ve subsh O n e can n eled by a disjoint. re a n o ti c u tr .> .i r s IJ n o c ) S K2 n 5 X(w, n ) J=O n , n>O, (w A 5 th e in1 K1 for which a d e r) . ) 4 W P re 1 -C j= O e for th e n (C P W n s tr u c ti o as a n easy exercis om th e above o c ic tr e fr ows t a. geom ave this (t h is foll . is th e n u m b e r in c o n s tr u c expanding (we le nformal o S> -c One c a n re t si e ions o a h n u w q it ts x dimens = S>,, a p G is duced m x p a n d in g .1)). T h u s ,! ! = s wer a n d u p p e r b o e is it if 3 lo However, a n d C o n d it io n (1 f dimension a n d es dorf · .) S> inequaliti .1 (i.e., th e Haus l to 13 a re equa T h e o re m s e t coincide a n d it o f th e lim
. F ig u r e 7
J1.>.i
Dimension of Cantor-like Sets and Symbolic Dynamics
153
16. Geometric Constructions with Rectangles; Non-coincidence of Box Dimension and Hausdorff Dimension of Sets
A crucial feature of Moran-like geometric constructions with stationary or non-stationary ratio coefficients is that they are, so to speak, isotrop~c, i.e., the ratio coefficients do not depend on the directions in the space. This is a very strong requirement and that is why the placement of the basic sets may be fairly arbitrary. A simple example of anisotropic geometric constructions is provided by constructions with rectangles. As in the case of Moran-like geometric constructions one still has a complete control over the sizes and shapes of the basic sets but needs two collections of ratio coefficients to control length and width of rectangles on the nth step. In this section we consider the simplest case when the ratio coefficients are constant and do not depend on the step of the construction. Even so, we will see that the Hausdorff and box dimensions of the limit set may depend on the placement of the basic sets and may not agree. This can happen even if a geometric construction is "most close" to a self-similar construction, i.e., it is given by finitely many affine maps (the so-called general Sierpinski carpets; discussed later in Section 16). In this section we present examples which illustrate how the equality between the Hausdorff dimension and box dimension can be de:,troyed. These examples are also a source for understanding some "pathological" properties of the pointwise dimension (see Section 25). A surprising phenomenon is that the Hausdorff dimension of the lirmt set for constructions with rectangles may also depend on some delicate number-theoretic properties of ratio coefficients corresponding to different directions. Geometric Constructions with Rectangles We call a symbolic geometric construction (CG1-CG3) on the plane modeled by a symbolic dynamical system (Q 1 u) a construction with rectangles if there exist 2p numbers~; and A;, i = 11 • • • ,p, 0 ..;~ we obtain that sin On satisfies the fol1owing equation: (an+l sin11n- 1/~) = 1 -- sin 2 On. 2
The formal roots of this equation are
. (}
SJD n
= 1
1 (1xa
+a
2(n+l)
n+l
-
±
1 - ;2
+ a2(n+l))
.
Since cos 8,. > 0 we have to choose the root with minus in front of the square root. The discriminant of this equation is non-negative if and only if lA., in I = [~2(n+l) + )..2(n+l)) 1/Z < _,i". This takes place if n is sufficiently large, i.e., IA;0 ;~+•! < }/'. One can also see that sin8n is asymptotically equivalent to (1/~ -1) a-(n+ll as n--+ oo. Since~< 1 there exist C 1 > 0 and C2 > 0 such that C1a-n < sin Bn < C 2 a-n for all sufficiently large n. One can now choose C 1 and C 2 to obtain in addition that C1a-" 0 is sufficiently large and n > m ?: no one has C1a-(m·H) [
1-a-t
1
_ a-(n-m)]
0 and k 2:: m ... Therefore, 1n(wk) = 1n(w) for n E N and k 2:: ffin. Using (16.1) we conclude that
i-y(wk) - ')'(w)i~i'Y(Wk) - 'Yn(wk) i+i'Yn(Wk) - 'Yn(w)i + i'Yn(w)- ')'(w)j< 2C(5.j},)-n
(16.3)
for n sufficiently large and k ;::: m ... Given e > 0, let us choose n > 0 such that 2C(5.j},)-n <e. Then I'Y(wk)- ')'(w)l < e fork 2:: m ... Since F is compact this gives the desired result. • Propositions 16.1 and 16.2 show that the basic sets of a symbolic geometric construction with rectangles cannot be spaced arbitrarily, but arc forced to be oriented to approach a continuous vector field. In [Bar 1), Barreira showed that the problem of computing the Hausdorff dimension and box dimension of the limit sets for geometric constructions with rectangles can be reduced to study constructions of a special kind where rectangles are aligned, i.e., the sides of length >:' for all n 2: 0 are parallel (see Figure 10). We present here two of his results in this direction. The proofs are based on Propositions 16.1 and 16.2 and are ormtted.
I I
I
t·
I
Figure 10. A CONSTRUCTION WITH ALIGNED RECTANGLES. Theorem 16.2. Given a geometnc construction with rectangles, modeled by a symbolic dynamical system (Q, a), there ensts a geometnc construction with altgned rectangles, modeled by (Q, a), such that dim8 F = dim 8 F and dimBF = dimBF, where F and F are the limit sets of the two constructions. We say that a geometric construction has exponentially large gaps if there exists o > 0 such that for each (i 0 i 1 •.. ) E Q and each n EN we have that
Dimension of Cantor-bke Sets and Symbolic Dynamics
157
dist(.6.; 0 ini• .6.io ink) 2': 8f!n whenever j =/= k and (io ... inj) and (io ... ink) are Q-admissible. With this extra hypothesis we can study the Hausdorff dimension of a geometric construction with rectangles by comparing it with an appropriate geometric construction with aligned rectangles. Theorem 16.3. Given a geometnc construction with rectangles and exponentially large gaps, modeled by a symbolic dynamical system (Q, u), there extsts a geometnc construction with aligned rectangles and exponentially large gaps, modeled by (Q, u), such t~at the map f: F_--+ F ts Lipschitz with Lipschitz inverse and dimH F = dimH F, where F and F are the limit sets. Non-coincidence of Hausdorff Dimension and Box Dimension We provide several special examples of geometric constructions with rectangles that illustrate some interesting phenomena and reveal the non-trivial structure of these constructions and the crucial difference between them and Moran-like geometric constructions with stationary ratio coefficients. We consider a geometric construction with rectangles in R2 which is generated by finitely many affine maps /!, ... , /p: S --+ S, where S = [0, 1] x [0, 1] is the unit square. Let F be the limit set. Each map !. can be written a.q fi(x, y) = T;(x, y) + b;, where T; is a linear contraction and b; is a two-vector. In [F2] (see also [F4]), Falconer proved that for almost all (b1 .•. bp) E R2P (in the sense of2p-dimensional Lebesgue measure), the Hausdorff dimension and the lower and upper box dimensions ofF coincide and the common value is completely determined by T1 , ... , Tp provided that liT; II < ~· On the other hand, as we mentioned above, the Hausdorff and box dimensions of the limit set for geometric constructions with rectangles may depend on the placement of the basic sets. They may not agree even in the case when the construction is generated by finitely many affine maps. The corresponding example is described by McMullen (see [Mu]; in [LG), Lalley and Gatzouras considered a more general version of his construction). Given integers e 2': m 2': 2 choose a set A conslSting of pairs of integers (i,j) wrth 0 ::; i < l and 0 ::; j < m. Denote by a the cardinality of A (clearly a ::; mn). Let J~c, k = 1, ... , a be affine maps given in the following way: if k enumerates the element (i,j) E A then fk(S) = S;;, where S = [0, 1) x [0, 1] and S;; = [~, ~] x [;;, ~] The affine maps fk generate a simple geometric construction with basic sets .6.io in = /; 0 o · · · o /;n (S) being (rn x m-n)rectangles. We allow some of the basic sets .6. 1, ... , .6.., on the first step of the construction to intersect each other by either a common vertex or a common edge (see Figure 11 where l = m = 4 and a = 6). The geometric description of this construction is the following: starting from the (l x m)-grid of the unit square S choose rectangles corresponding to (i, j) E A, then repeat this procedure in each chosen rectangle and so on. The limit set F for this construction is
F= {
(f~:. n=D
and is known as a general Sierphiski carpet (see [Mu]).
Chapter 5
158
Example 16.1. (1) dimH F = logm(l::j:~ t~og,m) ~ s, where t; is the number of those i for which (i, j) E A. (2) dimBF = dimBF = lo~ r + logf(~), where r is the number of those j for which (i,j) E A for some i. 1
Remarks. (1) For the geometric construction shown on Figure 11 we have t 0 = 1, t 1 = 2, t2 = 2, t3 = 1, and r = 4. (2) The Hausdorff and box dimensions of the limit set F agree if: a) l = m; b) the constants t; take on only one value other then zero. In the first case the construction is conformal self-similar and the common value for dimensions is logm a. Proof. Let X· r:;t -t F be the coding map given as follows:
~ .i::._) ~mn '
n=O
where kn = (in, in)· This map provides a symbolic representation of the limit set F by the (one-sided) shift on a symbols It is surjective but may fail to be injective because some points in F may have more than one representation. Define numbers bk, k = 0, ... , a- 1 in the following way. If k = (i,j) E A then bk is the number ofi' such that (i',j) EA. Note that by the definition of the numbers, 8 _
a-1 ""blog, m-1
m-L.,k
.
k=O
We define the Bernoulli measure on E;t by assigning probabilities blog,m-1
Pk = -'""---::--
m•
to each symbol k = 0, ... , a - 1. In other words, if Cko kn is a cylinder set then tt( Oko kn) = O~=O Pk,. Note that l::~=l Pk = 1. Let A be the measure on F which is the push forward of the measure J.l. (i.e., .\(A) = JJ-(x- 1 (A)) for any Borel subset A c F) We will show that dimH .\ ~ ,q Set q = [nlogt m] $ n. Given (n + 1)-tuple (ko ... k,.) with kt = (idt) E A, consider the set Rko kn C F of all points (x, y) for which 00
X= L.,fj• t=O
00
.,
""Zt
y=
.,
L~t' t=O
where i~ = it for t = 0, ... , q and J; = }t for t = 0, . , n The set Rko kn is "almost" a ball of radius m-n. More precisely, there are constants 0 1 > 0 and 02 > 0 independent of (ko ... kn) and a point (x,y) E Rko kn such that (16.4)
Dimension of Cantor-like Sets and Symbolic Dynamics
159
We will show that the sets R~oo ... kn comprise an "optimal" cover of J which we use to compute the Hausdorff dimension of the measure >... Note that each set R~oo. ,." can be decomposed into finitely many basic sets Llko kok~+> k~ n F, where k~ = (i~, jD and J~ = jt fort= q + 1, ... , n Moreover, the number of such sets is equal to b". ((x, y)) ~ s for A-almost every point (x, y) E F. Thus, s ~ dimH F. We now show that d.ImH F ~ s by constructing an efficient cover of F. Fixe> 0 and consider the collection 14. of those of sets Rko k,. for which r/Jko kn ~ m-•. These sets are disjoint and by Lemma 1 satisfy
Therefore, the number of such sets is bounded by mC•+•}n as ).(F) = 1 Note that any point (x, y) E F is covered by sets Rko k,. E 14. for infinitely many n since limn-too4>n((x, y)) ~ 1 > m-• (see Statement 1 of Lemma 2). Therefore, 'R.(N) = Un;::N'Rn is a cover ofF for any choice of N. Let us choose N so large that
It follows that
L R•o
"" E'R.(N)
(diantRko kJ(s+ 2 ~) =
L card'R.n m<s+ 2e)n ~ L m-en< e n;=:N
n;=:N
(here card denotes the cardinality of the corresponding set). This implies that dimH F ~ s + 2e and the desired result follows since c can be chosen arbitrarily small. In order to compute the box dimension of the limit set F let us choose n > 0 and set rn = m-n. Consider the finite cover ofF by sets Rko k,. and let Nn be the number of elements in this cover. It is easy to see that
where Ct > 0 and C2 > 0 are constants independent of n (recall that N(F, r) the least number of balls of radius r needed to cover the set F). Note that N,. is precisely the number of ways to choose sequences (it), t = 1, ... , q and (jt), t = 1, ... , n (recall that q = [n logt m]) such that a) (it,Jt) E A fort= 1, .. ,q, b) (i~, jt) E A for t = q + 1, ... , n and some choice of i~
ts
Chapter 5
162
It follows that N., = aqrn-q = (~)qrn (recall that a is the cardinality of A and r is the number of j such that (i, j) E A for some i). Therefore, di F ___ffiB
= n-+00lim lolg Nn ogr.,
F d lffiB = = --,--
= logm r
. logN(F,rn) 1liD
n-+oo
-
log r n
+ logm ~r n-+oo lim !!.. = logm r + logm ~. n r
This completes the proof of the statement.
•
Following Pesin and Weiss [PW1] we construct a more sophisticated example than in the previous section, which illustrates that all three characteristics - the Hausdorff dimension, the lower and upper box dimensions - may be distinct.
Example 16.2. There extsts a geometric construction with rectangles in the unite square S C JR2 , modeled by the full shift on two symbols (Et, u), for which _d 1 = _d2 = _d, X1 = "X2 =X, 0 < .d <X< ~, and the limit set F satisfies . log2 d1mHF= - -,, - 1og. _..!__ ')..n
3 -
I
I -n
-n
a
b
).
).
Figure 12. a) Vertical Stacking, b) Horizontal Stacking. We now compute the Hausdorff dimension and the lower and upper box dimensions of the limit set F. a) Calculation of Hausdorff Dimension.
Given e > 0, choose k > 0 such that An••+~ :5 e. Consider the cover ofF consisting of green rectangles for n = nak+l and blue rectangles for n = nak+2· Consider a green rectangle ~ 10 in•>+' . Dy construction the intersection A = ~; 0 in•>+• n F is contained in 2n••+•-naut small green rectangles corresponding to n = n 3 k+2. These rectangles are vertically aligned and have size 2 +. = xfna>+d+ :5 constAn"+' the (1 - f3k)2n••+• An•>+• X X'"+'. Since green rectangles in the construction of F arc each contained in a green square of size An•>+•. Now consider a blue rectangle ~; 0 in••+•. By our construction the intersection B = ~... in••+• n F is contained in 2n•>+a-na•+' small blue rectangles corresponding ton= n 3 k+3· They are vertically aligned and have size ~nsHa x -,r'••+a. Since -,r'ao+a :::; constAn••+• the f3k2n 3 •+'2n 3 '+ 2 -n3 '+' = {J~.,2n••+> blue rectangles in the construction of F are each contained in a blue square of size Ana>+• . The collection of green and blue squares comprises a cover g = {U;} ofF for which
r ..
E (diam U;)" :5 const ( (1 - fik)2nak+l (J2~nau• )" + fJk2n•>+• (J2Ans>+•)•) . U;EQ
The right-hand side of this inequality tends to 0 as k --+ oo if s > .!~~:~.' This implies that dimH F :5 ~~!g\. On the other hand, by Theorem 16.1, we know that dimH F ;::: .!~!:4 . Therefore, we conclude that log2 . ,. dtmnF= - 1og!l We now proceed with the box dimension. b) Calculation of Lower and Upper Box Dimensions.
Chapter 5
164
Choose e > 0. We wish to compute explicitly the number N(F, e) (the least number of ball of radius e needed to cover F). There exists a unique integer .,-n+l .,..,. n > 0 such that ~ < e ~ ~ . Denote by
A,= logN(~e). -log>.. We consider the following three cases: Case 1: n 3 ~;: ~ n < n 3 k+1· One can easily see that N(F,e) = 2" and hence,
A,= log2-· -log~
Case 2: na1:+1 ~ n < na/:+2· We have N(F,e) = Nblue(F,e) + Ngreen(F,e), where Nbtue(F,e) and Ngreen(F,e) are the numbers of e-balls in the optimal cover that have non-empty intersection with respectively blue and green rectangles at step n. It follows N(F,e) = f3~:2" + (1- f3~:)2"•w.
One can see that for all sufficiently large k (for which fJ1: ~ ~),
N(F, e)~ 2 ( 2(-y-n)nsHt2""a>+t+n-nsH2
= 2 (2 and
7
"••+'2"-"••+• + 2" 3 •+ 1 )
N(F,e) 2:: ~ (2(-y-n)ns>+t2"'"••+t+n-n 3Ho +2"•>+•) =
~ (2'Y1'••+'2"-"••+• + 2""+'). 2
One can easily check that the following inequality holds
provided nak+I < n < nak+2· This implies that lim A,.> "(log2 = "(log2 - - log~ -a log >..
n-+oo
Moreover, if n
+ 2ns•+t)
= na1:+1 then lim An= log2_. - log >..
n-+oo
Case 3: n 3k+2
~ n
< nak+3· We have
Dimension of Cantor-like Sets and Symbolic Dynamics
165
It is easy to see that for sufficiently large k (for which f3k ~ ~),
and N(F, e) Since
zn + 2'Yna•+•
;:::
~
( zh-cr)ns•+•z"'n3HI
;::: 2n
li ....ill
n3k+2
=
~ (zn + 2'Yn3k+l) .
we obtain that
n-+oo
provided
+ zn)
log2 A n;::: -1--,. -
< n < n3k+3· Moreover, if n
og~
= n3k+2 then
lim An= logz__ -log A
n-+oo
2 It follows that dimBF;::: - 101og & ,. Combining this with Theorem 16.1, we conclude ...
~~!s2>.. It also follows that dim8 F 2: "f _!~:~. As we have seen d" F ="'-log~. ~ • li A nsk+t =I ~ ab ove, k-+~ -log~· Thus, ...!!!1B
that dimBF
=
We consider another example of a simple geometric construction with rectangles in the plane generated by two affine maps which was introduced by Pollicott and Weiss (see [PoW]). It illustrates that the Hausdorff dimension of the limit set may depend on delicate number-theoretic properties of ratio coefficients while the box dimension is much more robust. Example 16.3 We begin with two disjoint rectangles ~lt ~ 2 C I in the unit square I= [0, 1] x [0, 1] given by ~; = [a;, a;+ A2] x J;, i = 1, 2, where 0 < a 1 ~ a2-< 1 and Jt, Jz are disjoint intervals in the vertical axis of the same length A1. We assume that 0 < At ~ A2 < 1 and At < ~- Consider the two affine maps h;: I-+ ~;, i = 1, 2 that contract the unit square by At in the vertical direction and by A2 in the horizontal direction. See Figure 13. These maps generate a simple self-similar geometric construction with rectangles in the plane with basic sets at step n
Let 'Irk, k = 1, 2 denote the projections of the unit square I onto the vertical side, for k = 1, and onto the horizontal side, for k = 2. Obviously, 7rt o h;(x, y) = a;+ Azx for any (x, y) E I and i = 1, 2. Hence, for any basic set ~; 0 in the left endpoint of the interval 7rt(~io ;J is given by n-t
7rt o h.o o h;1 o · ··oh.n(O,O) = L(at +ik+l(a2- at)) A~. k=O
Chapter 5
166
Taking the limit when n
11"2(F) =
-t
oo yields
{f)a1 +
i1c+l (az
-a I))
A~ :
0,
(i ih ... )
k=O
={ 1 ~\2 +df>k+tA~:
E{0, l}N}
(io,ih···)E{0,1}N}
(16.5)
k=O
c [
1~\ ' 1:2AJ = J,
a2-
where d = a1. Note that if A1 = A2 then the construction is a Moran simple geometric construction (CM1-CM5). It then follows from Theorem 13.1 that . F d lmH
.J:
=~
F
2
F log , • = -d. 1mB = - 1og"'2
(16.6)
There is a very special- degenerate- case when 11" 1 (~ 1 ) = 11" 1 (~ 2 ) (i.e., the rectangle ~ 1 lies directly above the rectangle ~ 2 ) It is easy to check that in this case (16.6) still holds. We now compute the Hausdorff dimension and lower and upper box dimensions of the limit set F of the construction assuming that At < A2 and d = a2- a1 > 0.
a) Calculation of Box Dimension. Lemma 1.
(1) dim8 F (2)
= dimBF ~f climB F. _..!2&1.. dimBF =
log>.,
{ - log ¥;/log At
i/O< A2:::; ~. if~ :::; ).2 < 1.
!·
Proof. We first consider the case 0 < A2 :::; By virtue of (16.5) the set 7rz(F) is aflinely equivalent to the standard Cantor set
after scaling by d = disjoint rectangles
a2 -
a1
> 0 and translating by
~·2 =
1 ~~. .
Consider two new
[a21 -dAz ~] - ).2 ' 1 - A2
>
0, e: > 0, and x E Y and choose sequences xk,l, such that xk,l -t x, Xk,2 -t x, rk, 1 -t r, rk,2 -t r, and
rk,l
and xk,2• rk,z
This imphes that (17 5) Moreover, since J.~-(8Dk) = 0 for all k, one can find K = K(e:) such that for any k ~ K(s) and i = 1,2, (17.6) It follows from (17.4), (17.5), and (17.6) that for any k 2: K(e) and n with n + m ~ n(x,e:, k),
~
N(x, D(x, r), n, m) < N(x, D(xk,2, rk,2), n, m) n+m n+m ~ J1-(D(xk,2• rk.2)) + e: ~ J.~-(D(x, r)) + 2e:, N(x, D(x, r), n,m) > N(x, D(xk,lt rk,2),n, m) n+m n+m ~ ~-t{D(xk,t. rk.d) - e 2: JL(D(x, r))- 2e.
0, m 2: 0
176
Chapter 6
Thus, for any x E Y and c > 0 there exists n 1 = n 1 (x,e:) 2: 0 such that for any n 2: 0, m 2: 0 with n + m 2: n1,
N(x,D(x,r),n,m) -J,£(D(x,r))'::::; Ze.
(17.7)
n+m
Since p. is ergodic and p.(D(x, r)) is a bounded Borel function over x E X we have by virtue of the' Birkhoff ergodic theorem that for J,£-almost every x E X, n-1
lim n--+oo
!_ Ltt(D(i(x),r)) = n
i=O
1
p.(D(y,r))dJ,£(y) = cp+(r).
X
This implies that for any e > 0 there exist a measurable set xJ~j c Y with p.(XJ.~) 2: 1 - e and a number n 2 = n 2(e) such that for any x E x$.~ and (17.8)
Moreover, for JL-almost every
X
E
xJ.~, (1)
lim n-+oo
N ( x, Xr,• , n, 0) _ (X{ll) -P. r 0 and any integer q ~ 2 consider the function
rp~(r) =
L
p,(D(x, r))q-l dp,(x),
where p, is an /-invariant Borel probability measure on X (recall that D(x, r) is the closed ball of radius r centered at x). The function tpt(r) is non-decreasing and hence may have only a finite or countable set of discontinuity points. It is clearly right-continuous. The following statement is proved by Pesin and Tempelman in [PT] and is an extension of Theorem 17.1 to correlation dinlensions of higher order. Theorem 17.2. If p, is an ergodic mMsure then for any integer q ~ 2 there extsts a set Z c X of full mrosure such that for any € > 0, R > 0, and any x E Zone can find N = N(x,e:,R) for which
with arbitmry n
~
N and 0
< r S R.
Remarks. (1) We stress that the lower and upper q-correlation dimensions do not depend either on the dynamical system f or on the point x for p,-almost every x (provided p, is ergodic). Instead, they are completely specified by the measure p,. This allows us to introduce the notion of q-correlation dimension for any finite Borel measure p, on a complete separable metric space X (see [PT]). Let Z c X be a Borel subset of pm.itive measure. Define the lower and upper q-correlation dimensions of the measure p, on Z for q = 2, 3, . . . by Corq(p,, Z)
=
_!__ dimqZ,
q- 1
Corq(p,, Z)
1-
= --dimqZ. q-1
For q = 2 these formulae define the lower and upper correlation dimensions of the measure p, on Z. We also define the (q, H)-correlation dimension of the measure p, on Z for q = 2, 3, ... by
Corq,H(JJ., Z)
= q ~ 1 dimq Z.
Multifractal Formalism
179
Consider the function cpq(r) = cpq(X,r) defined by (8.7). We stress that in (8.7) we use open balls ofradius r centered at points in X. A simple argument (which we leave to the reader) shows that
. logcpt(r) . logcpq(r) 11m = 1rm -:-=---7-'---'--7-
r-to
;::;a log(1/r)'
log(1/r)
-.-log cpt(r) -.-logcpq(r) hm =lim . log(1/r) r-+0 log(1/r)
r-+0
If p, is an invariant ergodic measure for a dynamical system follows that for J.t-almost every x E X, ~(x)
= ~(J.!, X),
liq(x)
f
acting on X it
= Corq(J.!, X)
(X can be also replaced by any set of full measure). These relations expose the "dimensional" nature of the notions of lower and upper q-correlation dimensions: they coincide respectively (up to the constant q~ 1 ) with the upper and lowerq-box dimensions of the space X for q = 2, 3, . . . (or any other set of full measure).
(2) Example 8.1 shows that the lower and upper q-correlation dimensions of p, may not coincide: namely, for any integer q 2: 2 there exists a Borel finite measure p. on I= [0, 1] for which
(and J.t is absolutely continuous with respect to the Lebesgue measure on I). One can check that J.t is an invariant measure for a continuous map f on I, where f(x) = J.!([O,x)). Thus, with respect to this map (17.13) for p.-almost. every :r E I Furthermore, using methods in [K) one can construct a diffeomorphism of a compact surface preserving an absolutely continuous Bernoulli measure with non-zero Lyapunov exponents for which (17.13) holds. (3) We describe a more general setup for introducing the lower and upper q-corrclation dimensions. Namely, given x,y EX, n > 0, r > 0, and an integer q 2: 2, we define the correlation sum of order q (specified by the points {Ji(x)} and {f;(y)}, i = 0, 1, ... , n) by Cq(x,y,n,r) =_.!__card {(i 1 .. . iq) E {0, 1, . . ,n}q:
nq
p(J 1i (x),ji•(y)) ::; r for any 0::; j, k ::; q}. Consider the direct product space Y = X x X. We call the quantities
_ .
.
1!Ill 1!ffi !!q (X, Y) - r-+0 n-+oo
JogGq(x,y,n,r) 1 1og (l/ r )
_ ( ) -. . JogCq(x,y,n,r) 11m 11m aq x, y = r-+0 n-+oo log ( 1I r )
the lower and upper q-correlation dimensions at the point (x,y) E Y. One can prove the following statement: let J.! be an f-invanant ergodic Borel
Chapter 6
180 probab-Jity measure and v = 1-1 x 1-'-i then for any integer q (x, y) E Y the limit 1un Cq(x,y,n,r) = cp:(X,r).
~ 2
and v-almost every
n-+oo
e:nsts.
(4} The functions cpq(X, r), q = 2,3, ... admit the following interpretation. Consider the direct product space (Yq, pq, vq), where Yq =X
X .•. X
X,
~
Vq
= ,_____...... 11- X •.. X 11-
q times
q times
=
= 2:::%"' 1 p(xk, Yk) for x = (xt. ... , xq), y (Yt. ... , yq)· Let A. { (x, ... , x) E Yq: x E X} be the diagonal. Then for any r > 0,
and pq(x, jj)
=
cpq(X, r) = vq(U(A., r)), where U(A., r) is the r-neighborhood of A.. (5) Let 1-1 be a Borel finite measure on am with bounded support. Following Sauer and Yorke [SY] we describe another approach to the notion of correlation dimension of p based on the potential theoretic method (see Section 7). Define the quantity (17.14) D(~J-) = sup{ s : I.(/1-) < oo }, where I.(JL) is the s-energy of p, (see (7.5)). The "correlation drmension" D(p) is interpreted as the supremum of those s for which the measure 11- has finite s-energy. Given a subset Z C Rm, one can compute the Hausdorff dimension of Z via the quantity D(p,). Namely, by the potential principle (see (7 6}), dimH Z
= sup{D(~J-) : ~-t(Z) = 1}.
(17.15)
"' A slight modification of the argument by Sauer and Yorke in (SY] shows that
where X is the support of JL. Indeed, set D = Cor2 (J-~, X). By the definition of the lower correlation dimension for any e > 0 there exists r 0 > 0 such that for any 0 < r ~ ro, cp2(X, r) S rD-•. Given 0 S s < D, set e = (D- s)/2 > 0. We have that 00
1 o
dcp2(X, r} = (" dcp2(X, r)
r•
Jo
+
r•
r "' < - n_!!_ e + constant
1, where
HP (J.£) = _1_lim logfx !J(B(x,r))q-I dJL(x), q- 1 r-->0 log(1/r)
==-q
-
( )
HPq 1-'
1
- . logfxJL(B(x,r))q-l dJL(x)
= q -1!~
log(1/r)
·
It follows from (8.10) that -
1-
HPq(lt) = --dimqX. q-1
{18.1)
Thus, the HP-spectrum for dimensions is a one-parameter family of characteristics of dimension type. for every q > 1 the quantities HPq(f£) and HPq(/-l) coincide (up to a normalizing factor q_: 1 ) with the lower and upper q-box dimensions of the set X respectively. Equalities (18.1) allow us to rewrite the definition of the H P-spectrum for dimensions in the following way: using (8.5) we obtain that
HP ( ) = _1_ltm log.:lq,..,(X,r) /-1 q- 1r-+0 log(1/r) '
==-q
HP (p) q
= _1_lim log.:lq,..,(X,r), q - 1 r-->0
log( 1/r)
Multifractal Formalism
183
where 1 > 1 is an arbitrary number and 6.q,'Y(X, r) is defined by {8.6). As Remark 1 in Section 17 shows for q = 2,3, ... the values HPq(P.) and HPq(P.) coincide with the lower and upper correlation dimensions of X, i.e, ~(JL, X) and Corq(JL, X). Following Pesin and Tempelman (PT] we introduce the modified H Pspectrum for dimensions specified by the measure JL as a one-parameter family of pairs of quantities (HPM q(JL), HPMq(p.)), q > 1, where 1 . lim!ogfzJL(B(x,r))q-l dp.(x) HPM ( p.) = - - 1 m 1 sup q - 1 o->0 z r->0 log(1/r) ' HPM ( ) 1 r rlogfzt-t(B(x,r))q-ld p.(x) q J1. = q- 1o~ s~p r~ log(1/r)
==-==q
and the supremum is taken over all sets Z c X with ft(Z) One can derive from Theorem 8.4 that for any q > 1, --
HPMq(p.)
~
1-6.
= -q-1 di1nqp.. 1
Thus, the modified HP-spectrum for dimensions is a one-parameter family of Caratheodory dimension characteristics specified by the measure 1-l· It follows from Theorem 9.2 that for any q > 1,
In particular, if the measure JL satisfies d_,.(x)
= dp(x) = d"'(x) then
(compare to (9.4)) . • The modified H P-spectrum for dimensions is completely specified by the equivalence class of JL. This was shown in (PT]. We present the corresponding result omitting the proof. Theorem 18.1. Let 11-1 and t-t2 be Borel measures on X. If these measures are equivalent then
It follows from the definitions that for q > 1
As Example 8.1 shows there exists a Borel finite measure p. on [O,p] for some > 0 such that
p
Chapter 6
184
for all 1 < q ::; Q, where Q > 0 is a given number (the reader can easily check .2=.!. that this holds provided /3 > a • ) .
R.enyi Spectrum for Dimensions Let U c am be an open domain. A finite or countable partition ~ = { 0;, i ?: 1} of U is called a (/3, r)-grid for some 0 < /3 < 1, r > 0 iffor any i?: 0 one can find a point x; E U such that
B(x;, {Jr)
c
C;
c B(x;, r).
The Euclidean spacer admits (/3, r)-grids for every 0 < /3 < 1 and r > 0. Let p, be a Borel finite measure on Rm and X the support of p,. We assume that X is contained in an open bounded domain U c am. Given numbers q ?: 0 and /3 > 0, we set .
D diDq
X
. log Aq(X, r) = r1liD , .... o log(1/r)
--.-- X -. logAq(X,r) DliDq = 1liD , r--+0 log(l/r)
where
Aq(X, r)
= inf { ~
2: p,(O;)q}
C,E~
and the infimum is taken over all {/3, r )-grids in U (compare to (8.6) ). We wish to compare the quantities DimqX, DimqX with the quantities dimqX, dimqX. The following result obtained by Guysinsky and Yaskolko [GY] establishes the coincidence of these quantities for q > 1 and demonstrates that one can lL~e grids instead of covers in the definition of q-dimension.
Theorem 18.2. If p. is a Borel finite measure on Rm with a compact support X then for any q > 1,
Let~= {C;, i?: 1} be a ({J,r)-grid. There are points {x,} such that B(x,,{Jr) C C; C B(x,,r). Therefore,
Proof.
::; 2: j
p.(B(x, 2r))q-l dp. =
i~l C;
This implies that Aq(X, r) ::; cpq(2r).
j p.(B(x, X
2r))q-l dJJ = cpq(2r).
Multifractal Formalism
185
We now prove the opposite inequality. Consider again a (.8, r)-grid {C;, 1} and choose points {x;} such that B(x;,,Br) C C; C B(x;,r). We have
i;:::
cpq(r)
= j ft(B(x,r))q-ld!J. =
~
L
j J.t(B(x,r))q-ldJL
L i~l
X
J
c,
~L
JL(B(x;,2r))q-ld1J
~1~
J.t(B(x;,2r))q.
~1
Therefore,
cpq(r) ~
L tt(B(x;, 2r})Q. i~l
Let us fix a ball B(x;, 2r) and consider the set D; = {x; : Cj n B(x;, 2r) :j= 0}. Note that D; C B(x;,3r) and that any two points x;.,xh ED; are at least 2.Br apart (recall that balls B(x., .Br) are disjoint). This implies that there exists N == N(.B) such that cardD; ~ N (we stress that N does not depend on r). This implies that any ball B(x;, 2r) can be covered by at most N clements C;~ of the grid. We will exploit the following well-known inequality: for any N > 0 and q > 1 there exists K = K(N, q) > 0 such that for any collection of numbers {a;, 0::; a; 1, i = 1, ... ,N},
s
We have that for any i
~
1, N
J.t(B(x;,2rW ~ K,Ltt(C;.)q. k=l
Now let us fix an element C; of the grid and consider the set A;== {x; : B(x;, 2r)n C;.:j= 0}. Again, one can see that A; C B(x;, 3r) and tha.t any two points Xj,, xh E A; are at least 2f3r apart. Therefore, we conclude that cardA; ~ N and hPnce cpq(r) S NKLp(C;)q t 1,
(18.3) Information Dimension Let~ be a finite partition of X. We define the entropy of the with respect to p, by
where Ce is an element of the partition
e.
partition~
Given a number r > 0, we set
H,..(r) = inf {H,..(O: diam~ :-:::; r}, ~
where diame = maxdiamC~ We define the lower and upper information dimensions of p, by
.
H,..(r)
-( )
I (p, ) = rhm ( / ), I ... o 1og 1 r
- . H,..(r)
/-L = r-+0 lim 1og (1/ r ) .
(18.4)
Obviously, l(p,) :-: :; f(p,). Proposition 18.1. (Y2) (1) f(p,) :"S; dimB/-L· (2) If !4(x) ;:::: d for p,-almost every x then l(p,);:::: d. Proof. Fix o > 0, r > 0 and let B 11 ••. ,BN, N = N(o, r) be r-balls that cover a set of measure;:::: 1- o. Set B1 = B 1 and fork= 2, 3, ... , N, k-l
Bk = Bk \
U B;.
i=l
The sets Bk arc disjoint and have diameter :-: :; r. Therefore, one can find measurable sets Ui, j = 1, ... , M(r) such that together with Bk they comprise a finite measurable partition~ of X of diameter 5 r. We recall the following inequality: if numbers 0 < t :-: :; 1 and 0 < Pk :-:::; 1, k = 1, ... , s are such that E~=l Pk = t then s
- :~:::>k logpk 5
-t logt + t logs.
k=l
It follows that
H,..(r) :-: :; H,..(O :-: :; log N(o, r) - o logo+ clog M(r)
187
M ultifractal Formalism
Therefore, -I() -r-logN(6,r) -:--6log6+6logM(r) -d. >:d-,-- X 1Im fJ. :::; r-+0 liD l l og (1/ r ) :::; lffiB/.L + u Ims . og (l/ r ) + r-+0 Since dimsX is finite the first statement follows by letting {J--+ 0. In order to prove the second statement choose a < d and {J > 0. There exists a set Z C X with 1-1(Z) 2: 1- {Janda number r 0 > 0 such that Jt(B(x, 1·)) ::=; r" for any 0 < r :::; r 0 and x E Z Fix r ::; r 0 and consider a finite partition E;, = {C1 , ... , Cn} of diameter::; r. Let B 1 be the collection of sets C; E E;, that have non-empty intersection with Z and B2 the collection of other sets C; E £;,. We have that J.t(Uc,EB 2 C,) ::; 8. Each C; E B1 is contained in a ball B(x;, r) centered at some point x; E C; and therefore, has /.£-measure::=; r". Thus, 1-28 H,.(E;,) 2: "'"" L /.L(Ci)log /.L(C;) 2: -"-(-r"'log r"')
= (1- 28)alog(1/r).
r
C,EB,
It follows that H,.(E;,) > (1- 28)a. log(1/r) -
•
This implies the second statement.
As an immediate consequence of Proposition 18.1 and Theorem 7.1 we obtain the following claim.
Proposition 18.2. Let /.L be a finite Borel measure on R"'. Assume that /.L is exact dimensional (see Section 7). Then l(fJ.) = l(J.£) = dimH J.l. It is conjectured that for "good" measures
l(J.t)
= q-+l,q>l lim !1q(J.L)
J(p,) =
lim Rq(J.L)
q-+l,q>l
=
lim HPq(fl),
q-H,q>l
= q-+l,q>l lim HP9 (~-t)
Below we will show that this conjecture holds for equilibrium measures corresponding to Holder continuous functions on conformal repellers for smooth expanding maps (see Section 21) and on basic sets for "conformal" axiom A diffeomorphisms (see Section 24).
/(a)-Spectrum for Dimensions We now introduce another dimension spectrum in order to describe the distribution of values of pointwise dimension of a measure. Let fJ. be a Borel finite measure on R"' and X the support of J.l· There is the multifractal decomposition of X associated with the pointwise dimension of f.t:
Chapter 6
188 where
X= {x EX: fl,.(x) < d"(x)} is the irregular part and the level sets Xa are defined by -
def
Xa = {x EX: .r!.,.(x) = d,.(x) = d,.(x) =a}.
(18.5)
In order to characterize this multifractal decomposition quantitatively we introduce the dimension spectrum for pointwise dimensions of the measure 1-' or fl'(a)-spectrum (for dimensions) by
We first consider those values a for which f.t(Xa) > 0. Theorem 18.3. If 1-'(Xa) > 0 then dimH Xa = a. Proof. The statement immediately follows from Theorems 71 and 7.2.
•
If ~t(Xa) = 0 the set Xa may still have positive Hausdorff dimension and hence be "observable" from a physical point of view. Thus, the !,.(a)-spectrum can be used to describe the fine-scale structure of the measure It provided the function d"(x) exists on an "observable" set of points in X. An invariant ergodic measure f.t supported on a repeller of a smooth expanding map (see definition in Section 20) or on a hyperbolic set of an axiom A diffeomorphism (see definition in Section 22) can be shown to be exact dimensional (see Theorems 21.2 and 24.2). This means that the function dl'(x) exists and is constant on a set of full measure. Therefore, f.t(Xa) = 0 for all values a hut one. A pnon, it is not clear at all why the sets X a are not empty for all values a except this special one. Even if these sets are not empty, it seems unclear from a point of view of the classical measure theory that the function f I' (a) would behave "nicely" and provide meaningful information about the measure I.J. Furthermore, Barreira and Schmeling [BS] showed that for any equilibrium measure (corresponding to a HOlder continuous function) for a smooth expanding map or for an axiom A diffeomorphism the set X IS obseroable; moreover, it has full Hausdorff dimension (see Appendix IV for details; some very special cases should be excluded). Despite this it was conjectured in [HJKPS] that for "good" measures, which are invariant under dynamical systems, the function f"(a) is correctly defined on some interval [a~, a2], real analytic and convex. This wa.s supported by a computer simulation of some dynamical systems: a typical "computer made" graph of the function f"(a) is drawn on Figure 17b in Chapter 7. Furthermore, a strong connection between the HP-spectrum and /"(a)spectrum was discovered to support an important role that these spectra play in numerical study of dynamical systems: heuristic arguments (based on an analogy with statistical mechanics; see introduction to this chapter) show that the HP-spectrum for dimensions (multiplied by q- 1} and the /"(a)-spectrum for dimensions form a Legendre transform pair.
Multifractal Formalism
189
The rigorous mathematical study of the above multifractal decomposition and f 14 (a)-spectrum is based upon constructing a one-parameter family of equilibrium measures Va, a E [a1. a2] which are supported on X 0 (i.e., li0 (Xa) = 1) and have full dimension (i.e., dimE v0 = dimH X,.). We will construct such families of measures for multifractal decomposition generated by equilibrium measures invariant under conformal dynamical systems of hyperbolic type (see Chapter 7) We will also reveal extremely complicated "bizarre" structure of this multifractal decomposition: each set X 0 is everywhere dense in X and so is the set X. This also shows that the Hausdorff dimension in the definition of the f 14 (a)-spectrum cannot be replaced by the box dimension. Finally, we will observe that the !,..(a)-spectrum is complete, i.e., for any a outside the interval [at, a2], the set X a is empty.
19. Multifractal Analysis of Gibbs Measures on Limit Sets of Geometric Constructions
In this section we will show how to compute the Hentschel-Procaccia spectrum, Renyi spectrum, and !(a)-spectrum for dimensions of equilibrium measures supported on the limit sets of some geometric constructions (CG1-CG3) in JRm (see Section 13). We will also undertake a complete multifractal analysis of these measures. Continuous Expanding Maps Let X C lR"' be a compact set. We say that a continuous map f: X -+ X is expanding iff is a local homeomorphism and f satisfies Condition (15.9), i.e., there exist constants b :2: a > 1 and ro > 0 such that
B(f(x),ar)
C
J(B(x,r))
C
B(J(x),br)
(19.1)
for every x E X and 0 < r < ro. Without loss of generality we may assume that for any x E X the map I rem:ricted to the ball B(x, ro) is a homeomorphism. Note that a continuous expanding expanding map is (locally) bi-Lipsch.ttz. We recall that a Markov partition for an expanding map f is a finite cover R = {Rb ... , Rp} of X by elements (called rectangles) such that: (a) each rectangle R, is the closure of its interior intR,; (b) intR, n intRi = 0 unless i = j; (c) each f(R,) is a union of rectangles Rj. An expanding map has Markov partitions of arbitrary small diameters. Let us fix a Markov partition R. It generates a symbolic model of the map I by a subshift of finite type (:E1, u), where A = (a;;) is the transfer matrix of the Markov partition, namely, a;j = 1 if intR, n f- 1 (intRj) f= 0 and a;j = 0 otherwise. This gives the coding map x: :E1 -+ X such that
x(w) =
nh(R,n), for n~O
w = (ioil ... )
(19.2)
190
Chapter 6
(where his an appropriate branch of 1-n) and the following diagram ~1~E1
x~x is commutative. Under the coding map the cylinder sets Cio .in to the basic sets in X generated by the Markov partition
R.o in
= R.o n r 1 (R.l n r
1
(· ..
n
r
1
C
E1 correspond
(R,J ... )).
(19 3)
The map x is Holder continuous and injective on the set of points whose trajectories never hit the boundary of any element of the Markov partition. The pullback by x of any Holder continuous function on X is a Holder continuous function on E1. Let f be a continuous expanding map of a compact set X c R"'. We obtain effective estimates for the Hausdorff dimension and box dimension of X following the approach suggested by Barreirain (Bar2]. Let n = {R1 , .•. ,Rp} be a Markov partition of X of a small diameter c (which should be less than the number r 0 in (19.1)) and (E1, o-) the symbolic representation of X by a subshift of finite type generated by n. The Markov partition allows one to view X as the basic set for a geometric construction (modeled by the subshift of finite type) whose basic sets are defined by (19.3) and whose induced map is f. We will extend the approach, developed in Section 15 for expanding induced maps, to arbitrary continuous expanding maps, and we will apply the non-additive version of thermodynamic formalism to compute the Hausdorff dimension and box dimension of X. Fix a number k > 0 For each w = (ioi 1 . ) E E1 and n 2: 1 define numbers A (w n) ~ '
X (w k
= inf { lir(x) - r(y)li} llx-yjj
n) =sup { llr(x)'
rCY)II}
llx- Yll
'
(19.4)
'
where the infimum and the supremum are taken over all distinct x, y E X n R,0 '"+> (compare to (15.7)). Note that since f is expanding Condition (15.8) holds. Define two sequences of functions on E1 5f(k)
= {i,kl(w) = -logAk(w,n)},
~(k) = {~~kl(w) = -log_&(w,n)}. (19.5)
Consider Bowen's equations (19.6)
One can show that for any sufficiently large k, any wE ~1. and n 2: 1, an:::; _&(w, n):::; Ak(w, n) :::; bn (see the proof of Theorem 15 5). Thus, Condition (A2.23) (see Appendix II) holds. By Theorem A2 6 this ensures the existence of unique roots of Bowen's equations (19.6). We denote these roots by §.(k) and s-(k) respectively. Clearly, lk) :::; s 0 and k > 0 such that for each w E E~ and n ~ 0 Condition (15.11) holds. The following result is similar to Theorem 15.5. Proposition 19.2. [Bar2] Assume that f is topologically mrn-ng. Then for any open set U C X and all sufficiently large k,
and s is a unique number satisfying (diam(X nR; 0 i.))' = 0.
lim .!.log
n-+oon
(io
in) E!·a.dmissibJe
Weakly-conformal Maps We say that a map f of a compact set X is weakly-conformal if there exist a Holder continuous function a(x) with Ja(x)J > 1 on X and positive constants C1, 0 2 , and ro such that for any two points x, y E X and any integer n ~ 0 we have: if p(fk(x), fk(y)) :::; r0 for all k = 0, 1, ... n then
n n
Ctp(x,y)
Ja(f"(x))J- 1
:::;
p(f"(x), J"(y))
k=O
n
(19.7)
n
:::; C2p(x, y)
Ja(J"(x))l-
1
.
k=O
Obviously, a weakly-conformal map is continuous, expanding, and quasi-conformal and Proposition 19 2 applies We now discuss the diametrically regular property of equilibrium measures corresponding to Holder continuous functions for weakly-conformal maps (see Condition (8.15)). If f were a smooth map this property would follow from Proposition 21.4. The proof in the general case is a modification of arguments in the proof of this proposition and is omitted. Proposition 19.3. Let f be a weakly-conformal map of a compact set X and
0 r
where the infimum is taken over all finite covers mdms r In particular, for q > 1,
,
9r of X by open balls of
T(q) - - = HPq(v) = HPq(v) = Eq(v) = Rq(v). 1-q
The following statement is an immediate corollary of Theorem 19.1 (see Statement 1). Proposition 19.4. Any equilibnttm measure corresponding to a Holder continuous function for a topologically mmng weakly-conformal map f on X is exact dimensional. In fact, this result holds for an arbitrary (not necessarily equilibrium) ergodic measure for a weakly-conformal map. The proof is quite similar to the proof of Theorem 21.3 (which deals with the smooth case).
Induced Maps for Geometric Construction s (CG1-CG3) : Multifractal Analysis of Equilibrium Measures Consider a geometric construction (CG1-CG3) in IRm (see Section 13) and assume that it is modeled by a transitive subshift of finite type (:E1, u). Let F be the limit set. Since we require the separation condition (CG3) the cod.ing map x is a 1 homeomorphism and we can consider the induced map G = x o u o x- on the we constructions geometric the limit set F. It is a local homeomorphism since a builds one if Moreover, (:E1,u). type finite of subshift a by modeled consider are geometric construction modeled by an arbitrary symbolic system (Q, u) with the expanding induced map on the limit set then uiQ must be topologically conjugate to a subshift of finite type. This follows from a result of Parry [Pa]. Therefore, the induced map G is expanding if and only if it satisfies Condition (15.9). Note that the placement of basic sets of a geometric construction, whose induced map is expanding, cannot be arbitrary and must satisfy the following special property: there are constants C1 > 0 and C2 > 0 such that for every x E F and any r > 0 there exists n > 0 and basic sets C,(l), ... , c,(m) (with m = m(x, r)) for which B(x,C1r)nFc
U
L\(!:lnFcB( x,C2 r)nF.
l:S;k~m
We assume that the induced map is weakly-conformal. Note that in this case it is also quasi-conformal (see Cond.ition (15.11)). We conjecture that one can
Chapter 6
194
build a geometnc construction modeled by the full shift with disjoint basic sets such that the induced map is quasi-conformal but not weakly-conformal Theorem 19.1 can be used to effect the complete multifractal analysis of equilibrium measures, corresponding to HOlder continuous functions, supported on the limit sets of geometric constructions (CG1-CG3) with weakly-conformal induced map. We apply this result to a special class of self-similar geometric constructions (see Section 13). Recall that this means that the basic sets ~.0 • >n are given by where h11 ... ,l~,p:D---+ Dare conformal affine maps, i.e., llhi(x)- ht(Y)II = >.illx- yl! with 0 < >.; < 1 and x, y ED (the unit ball in lRm ). Assuming that the basic sets~ •• i = 1, .. ,pare disjoint, one can easily see that the induced map G on the limit set F is weakly-conformal (with a(x) = >.;;; 1, where x(x) = (ioi1 ... )). Thus, Theorem 19.1 applies. For Bernoulli measures this result was obtained by several authors who used various methods (see, for example, [CM], [EM], [F1], (0], (Ri]) We describe a more general cla::~s of geometric constructions to which Theorem 19.1 applies Namely, consider geometric constructions build up by p sequences of hi-Lipschitz contraction maps h~n): D---+ D such that
and for any x, y ED,
4ln) dist(x, y)::::; dist(h~n)(x), h~n}(y)) ::::; x~n) dist(x, y), where 0 < 4lnl : : ; X~n) < 1 (see [PW2]). We assume that the following asymptotic estimates hold: there exist 0
., < 1 such that
(19.8) One can check that the induced map G is weakly-conformal (with a(x) = >.i;; 1 , where x(x) = (i 0 i 1 ... )) and Theorem 19.1 applies Example 25.2 shows that there exists a geometric construction produced by three sequences of hi-Lipschitz contraction maps which do not satisfy the asymptotic estimates (19.8). Although the ba::~ic sets at each step of the construction are disjoint it does not admit the multifractal analysis described by Theorem 19.1 (otherwise, by Proposition 19 3 any equilibrium measure corresponding to a Holder continuous function on the llmlt set F of this construction would be exact dimensional which is false for the geometric construction described in this example). It is still an open problem in dimension theory whether one can effect the complete multifractal analysis of Gibbs measures supported on the hmit set of a Moran geometric construction (CM1-CM5) modeled by a transitive subshift
Multifractal Formalism
195
of finite type (for some results in this direction see [PW2, LN]). Notice that by Theorem 15.4, the pointwise dimension of such a measure exists (and is constant) almost everywhere. Remarks. (1) One can show that if 11 = m.>. then T(q) = (1 - q)s (thus, T(q) is a linear function) and f.,(o:) is the ¢-function (i.e., f.,(8) = 8 and f,.(o:) = 0 for all a 1- s; see Remark 1 in Section 21; thls case was studied by Lopes [Lo]). (2) The graphs offunctions T(q) and f(o:) are shown on Figures 17a and 17b in Chapter 7. Note that the function f(a) is defined on the interval (o: 11 o: 2 ], where a1 =- lim T'(q), az =- lim T'(q). q4+oo
q-4-oo
It attains its maximal value 8 at o: = o:(O). Furthermore, f(a(l)) = a(l) is the common value of the lower and upper information dimensions of v (see Section 21) and /'(o:(l)) = 1. We also note that the /(a)-spectrum is complete, i.e., for any o: outside the interval [o:1, 0:2] the corresponding set X a is empty (see Section 21). (3) Consider again a self-similar geometric construction modeled by the full shift u and assume that v is the Bernoulli measure defined by the vector (Pl , ... , Pr), where 0 < Pk < 1 and 2:~= 1 Pk = 1. It is easily seen from Theorem A2 8 (see Appendix II) that PEt {log(.Afo(q)P~0 l) is equivalent to
r
:L "'r(q)Pk = 1. k=l
=0
Chapter 7
Dimension of Sets and Measures Invariant under Hyperbolic Systems
In this Chapter we study the Hausdorff dimension and box dimension of sets invariant under smooth dynamical systems of hyperbolic type. This includes repellers for expanding maps and basic sets for Axiom A diffeomorphisms. The reader who is not quite familiar with these notions, can find all the necessary definitions and brief description of basic results relevant to our study in this chapter. For more complete information we refer the reader to (KH]. We recover two major results in the area: Ruelle's formula for the Hausdorff dimension of conformal repellers (see Theorem 20.1) and Manning and McCluskey's formula for the Hausdorff dimension of two-dimensional locally maximal hyperbolic sets (see Theorem 22.2). Our approach differs from the original ones and is a manifestation of our general Caratheodory construction (see Chapter 1) and the dimension interpretation of the thermodynamic formalism: it systematically exploits the "dimension" definition of the topological pressure described in Chapter 4 and Appendix II. This unifies and simplifies proofs and reveals a non-trivial relation between the topological pressure of some special functions on the invariant set and the Hausdorff dimension of this set. Furthermore, we use Markov partitions to lay down a deep analogy between conformal repellers (as well as two-dimensional locally maximal hyperbolic sets) and the limit sets for Moran-like geometric constructions We then apply methods developed in Chapter 5 (see Theorem 13.1} to study dimension of these invariant sets. In particular, this allows us to strengthen the results of Ruelle and of Manning and McCluskey by including the box dimension of repellers and hyperbolic sets into consideration. We also cover the case of multidimensional conformal hyperbolic sets. The crucial feature of the dynamics, to which our methods can be applied, is its conformality. In the non-conformal case (multidimensional non-conformal repellers and hyperbolic sets) the approach, based upon the non-additive version of the thermodynamic formalism, allows us to obtain sharp dimension estimates. We stress that in this case the Hausdorff dimension and box dimension may not agree. We describe the most famous examples of repellers (including hyperbolic Julia sets, repellers for one-dimensional Markov maps, and limit sets for reflection groups; see Section 20) and hyperbolic sets (including Smale horseshoes and Smale--Williams solenoids, see Section 23). We also provide a brief exposition of
196
Dimension of Sets and Measures Invariant under Hyperbohc Systems
197
recent results on the Hausdorff dimenaion of a class of three-dimensional solenoids by Bothe and on the Hausdorff dimenaion and box dimension of attractors for generalized baker's transformations by Alexander and Yorke, by Falconer, and by Simon (see Section 23). These results illustrate some new methods of study that have been recently developed, as well as reveal some obstructions in studying the Hausdorff dimenaion in non-conformal and multidirnenaional cases. A significant part of the chapter is devoted to the recent innovation in the dimension theory of dynamical systems - the multifractal analysis of equilibrium measures (corresponding to Holder continuous functiona) supported on conformal repellers (see Theorem 211) or two-dimensional locally maximal hyperbolic sets (see Theorem 24.1; in fact, we cover more general multidimensional conformal hyperbolic sets). The first rigorous multifractal analysis of measures invariant under smooth dynamical systems with hyperbolic behavior was carried out by Collet, Lebowitz, and Porzio in [CLP] for a special class of measures invariant under some one-dimensional Markov maps. Lopes [Lo] studied the measure of maximal entropy for a hyperbolic Julia set. Pesin and Weiss [PW2] effected a complete multifractal analysis of equilibrium measures for conformal repellers and conformal Axiom A diffeomorphisms. In this chapter we follow their approach. Simpelaere [Si] used another approach, which is based on large deviation theory, to effect a multifractal analysis of equilibrium measures for Axiom A surface d.iffeomorphisms. There are two main by-products of our multifractal analysis of equilibrium measures on conformal repellers and hyperbolic sets. The first one is that these measures are exact dimensional (see Theorems 21.3 and 24.2; in Chapter 8 we extend these results to arbitrary hyperbolic measures). The second one is the complete description of the dimenaion spectrum for Lyapunov exponents for expanding maps on conformal repellers (see Theorem 21.4) and diffeomorphisms on locally maximal conformal hyperbolic sets (see Theorem 24.3 and also Appendix IV). This spectrum provides important additional information on the deviation of Lyapunov exponents from the mean value given by the Multiplicative Ergodic Theorem.
20. Hausdorff Dimension and Box Dimension of Conformal Repellers for Smooth Expanding Maps Repellers for Smooth Expanding Maps Let M be a smooth Riemannian manifold and f: M -4 M a Cl+"-map. Let J be a compact subset of M such that f(J) = J We say that f is expanding on J and J is a repeller if (a) there exist C > 0 and>.> 1 such that lldf;,'vll 2: CN'IIvll for all x E J, v E T.,M, and n 2: 1 (with respect to a Riemannian metric on M); (b) there exists an open neighborhood V of J (called a basin) such that J = {x E V : r(x) E V for all n 2: 0} Obviously, f is a local homeomorphism, i.e., there exists ro > 0 such that for every x E J the map JIB(x, r 0 ) is a homeomorphism onto its image. Thus, f is expanding as a continuous map (see Section 19 and Condition (19.1)).
198
Chapter 7
We recall some facts about expanding maps. A point x E M is called non-wandermg if for each neighborhood U of x there exists n 2 1 such that r(U) n U =/= 0. We denote by fl(f) the set of all non-wandering points of f. It is a closed /-invariant set. The Spectral Decomposition Theorem claims that the set fl(f) can be decomposed into finitely many disjoint closed /-invariant subsets, fl(J) = h U · · · U Jm, such that f I J; is topologically transitive. Moreover, for each i there exist a number n; and a set A; c J; such that the sets f"'(A.) are disjoint for 0:::; k < n;, their union is the set J;, fn•(A;) =A;, and the map I A; is topologically mixing (see [KH] for more details). From now on we will assume that the map f is topologically mixing. This is just a technical assumption that will allow us to simplify proofs. In view of the Spectral Decomposition Theorem our results can be easily extended to the general case (with some obvious modifications). An expanding map f has Markov partitions of arbitrarily small diameter (see definition of the Markov partition in Section 19; see also [R2)). Let R = {Rll ... , R,} be a Markov partition for f. It generates a symbolic model of the repeller by a subshift of finite type (I:~, 0 There enst positive constants L 1 = L 1 ('1f;) and L 2 = L 2 ('1/J) such that for any (n + 1)-tuple (io ... in) and any x, y E Rio in,
rr n
Ll :::;
'1/J(!k(x))
k=O '1/J(f"(y))
:::; L2.
{2) Let '1/J be a Holder continuous function on J such that '1/J(x) 2 c > 0. There enst positive constants L 1 = L 1 ('1f;) and L2 = L2('1/J) such that for any n > 0, any branch h of and any points x E J, y E h(B(x, ro))
rn,
Statement 1 holds. (3) There exist positive constants L 1 and L 2 such that for any (n +I)-tuple (io .. in) and any x,y E R 00 in'
< IJacr(x)l Ll- IJacfn(y)l :::; L2, where Jacfn denotes the Jacobian of fn.
Dimension of Sets and Measures Invariant under Hyperbolic Systems
Proof. Let {3 > 0 be the Holder exponent and C1 Then
ll'I/J(f:(x)) 5 ll (1 + k=O (y)) By the expanding property we find that k=O '1/J(f
199
> 0 the Holder constant of '1/J.
d
lfk(x)- fk(y)I.B).
l!k(x)- !k(y)l 5 C2>.k-nlr(x)- r(y)l, where Cz > 0 is a constant. Therefore,
fi'I/JCJ:(x)) 5
k=O '1/J(f
(y))
n
(1+CICf(>.")k).
k=O
This implies the upper bound. The lower bound follows by interchanging :E andy. This completes the proof of the first statement. The second statement can be proved in a similar fashion The third statement follows by applying the first one to the function t/J(x) = Jac f(x) which is Holder continuous since f is of class
•
c~+a.
A Markov partition n = {Rb ... ,Rp} allows one to set up a complete analogy between limit sets of geometric constructions (CGI-CG2) and repellers of expanding maps by considering the sets R.o in as basic sets. Namely,
J=
n u R.o n::O:O {io
int
in)
where the union is taken over all admissible (n + 1)-tuples (io ... in). By the Markov property every basic set R.o. ;n = h(Rin) n R; 0 for some branch h of ~-n+l,
A smooth map f: M -7 M is called conformal if for each x E X we have df., = a(x) Isom,, where Isom, denotes an isometry of T.,M and a(x) is a scalar. A smooth conformal map f is expanding if la(x)l > 1 for every point x E M. The repeller J for a conformal expanding map is called a conformal repeller. Note that a smooth conformal expanding map is weakly conformal (as a continuous expanding map; see Condition 19.7). The converse is not true in general: there exists a C 00 -map which is not conformal in the above sense but is weakly-conformal (see [Bar2]). Conformal repellers can be viewed as limit sets for Moran-like geometric constructions with non-stationary ratio coefficients since the basic sets R.o in satisfy Condition (B2). Proposition 20.2. (1) Each basic set Rio ;. contains a ball of radius r., 0 in and is contained in a ball of mdi'IJ.S 1';0 in· (2) There eX?-st positive constants K 1 and K2 such that for every basic set R; 0 • in and every X E R.o. in , n
K1fl la(Ji(x))l-l 5 !:; 0 i=O
n
in
5 7'; 0 in 5 K2fl la(!l(x))l- 1. j=O
(20.1)
200
Chapter 7
Proof. Since f is conformal and expanding on J we have for every x
E J,
n
lld/;'11
= ll ia(!i(x))i = IJacr(x)l. )=0
This fact and the third statement of Proposition 20.1 imply that for every x E
Rzo
1n'
. = d1amR,n
(maxyER.n IJach(y)l) I ( nc ))I IJach(/"(x))J Jach f x ~
c1 J=O Tin I ( i( ))1-1 af x ,
where h is some branch of 1-n and C 1 > 0 is a constant. Since each Rj is the closure of its interior we have diamRio in (x) ~ diamR;" .
= diam.R.n
X
min 1Jdh11 11 = diam.R." x min IJac h(y) I
yERin.
yER,n.
(min11eR,n IJach(y)l) n IJach(tn(x))J IJach(f (x))J ~
Tin c2 j=O
j
Ja(f (x))i
-1
'
where C 2 > 0 is a constant. This completes the proof of Proposition 20.2. • We use the analogy with geometric constructions to define a Moran cover of the conformal repeller. It allows us to build up an "optimal" cover for computing the Hausdorff dimension and lower and upper box dimensions of the repeller. Given r > 0 and a point w E ~1. let n(w) denote the unique positive integer such that n(w)
n(w)+l
k=O
k=O
n Ja(x(uk(w)))l-1 > r, n la(x(uk(w)))i-l ~ r.
(20 2)
It is easy to see that n(w) -t oo as r -t 0 uniformly in w. Fix w E ~1 and consider the cylinder set Cio ;n<w> C ~1. We have that wE C;0 ;n<wJ" Furthermore, if w' E C; 0 •• in<w> and n(w') ::; n(w) then
Let C(w) be the largest cylinder set containing w with the property that C(w) = C; 0 ;n. These sets comprise a disjoint cover of ~1 which we denote by ll,. and call a Moran cover. The sets R(J) = x(C(il), j = 1, . . , Nr may overlap along their boundaries. They comprise a cover of J (which we will
Dimension of Sets and Measures Invariant under Hyperbolic Systems
201
denote by the same symbolU,. if it does not cause any confusion). We have that Xj E J. Let Q C E! be a (not necessarily invariant) subset. One can repeat the above arguments to construct a Moran cover of the set Q. It consists of cylinder , Nr for which there exist points Wj E Q such that C(j) = sets c and the intersection c 1). One says that R is a hyperbolic rational map (and that J is a hyperbolic Julia set) if the map R is expanding on J (i e., it satisfies Conditions (1)-(2) in the definition of smooth expanding map with respect to the spherical metric on C). It is known that the map z t-+ z 2 + c is hyperbolic provided lei < (see Figure 15). It is conjectured that there is a dense set of hyperbolic quadratic maps. (2) One-Dimensional Markov Maps [RaJ. Assume that there eXIsts a finite family of disjoint closed intervals Jr, [z,. Ip c I and a map f: Ui I; -t I such that
l
(a) for every j, there is a subset P ~ P(j) of indices with /(I;)== UkEPh (mod 0); (b) for every x E Ujintl;, the derivative off exists and satisfies I f'(x) I ~ ar for some fixed ar > 0; (c) there exists A > 1 and no > 0 sucli that if fm(x) E U; intij, for all 0 :S m:::; no- 1 then J(/" 0 )'(x)J ~A. Let J = {x E I : r(x) E U~=I I; for all n E N}. The set J is a repeller for the map f. It is conformal because the domain off is one-dimensional (see Figure 16).
202
Chapter 7
a
b
Figure 15. THE
BOUNDARIES OF THESE "BLACK SPOTS" ARE JULIA SETS FOR THE POLYNOMIAL z 2 c WITH
a) c = -
fo- + ~i and b) c = -fo -
+
~i.
0
Figure 16. A
ONE-DIMENSIONAL MARKOV MAP.
(3) Conformal Toral Endomorphisms. This is a map of multidimensional torus defined by a diagonal matrix (k, ... , k), where k is an integer and JkJ > 1.
Dimension of Sets and Measures Invariant under Hyperbolic Systems
203
(4) Induced Maps. Let h1, ... ,hp:D --t D be conformal affine maps of the unit ball D in !Rm. Assume that the sets h;(D) are disjoint. Consider the self-similar construction generated by these maps and modeled by the full shift (or a subshift of finite type; see Section 13) Define the map G on U; inth;(D) by G(x) = hi 1 (x) if x E inth;(D). Clearly, G is a smooth conformal expanding map and the repeller for G is the limit set of the geometric construction (i.e., G is a smooth extension of the induced map). Similar result holds for the induced map on the limit set of the geometric construction generated by reftection groups (see Section 15). Hausdorff Dimension and Box Dimension of Conformal Repellers
We compute the Hausdorff dimension and box dimension of a conformal repeller. Let f: M --t M be a Cl+"'-conformal expanding map with a conformal repeller J. Let m be a unique equilibrium measure corresponding to the Holder continuous function -slogja(x)l on M, where sis the unique root of Bowen's equation P;( -slog Ia/) = 0 (see Appendix II). Theorem 20.1. (1) dimH J = dim 8 J
= dimsJ = s; moreover s= J;logia(x)idm(x)"
(2) The a-Hausdorff measure of J is positive and finite; moreover, it is equivalent to the measure m. (3) s = dimH m, in other words, the measw-e m is an invariant measure of full Caratheodory dimension {see Section 5).
In [R2], Ruelle proved that the Hausdorff dimension of the conformal repeller J of a Cl+"-map is given by the root of Bowen's equation PJ( -slog Ia I) = 0. He also showed that the a-Hausdorff measure of J is positive and finite. In [F4], Falconer showed that the Hausdorff and box dimensions of J coincide. One can derive Theorem 20.1 from a more general result for continuous weaklyconformal expanding maps (see [Bar2] and Section 19). In particular, this shows that Theorem 20.1 holds for C 1-conformal expanding maps. This result was also established by Gatzouras and Peres [GaP]; see also Takens [T2] for a particular case. We provide here an independent and straightforward proof of Theorem 20.1 which is similar to the proof of Theorem 13.1 (where we dealt with limit sets of geometric constructions (CPW1--CPW4)) • Proof of the theorem. Set d = dimH J. We first show that s:::; d. FixE> 0. By the definition of the Hausdorff dimension there exists a number r > 0 and a cover of J by balls Bt, l = 1, 2, ... of radius rt :::; r such that
(20.3)
204
Chapter 7
Let 'R. = {Rl> ... ,Rp} be a Markov partition for f. For every f > 0 consider a Moran cover U,., of J and choose those basic sets from the cover that intersect Note that R(J) Bt. Denote them by Rt(I) , ... , R(m(t)) l . t = R ;0 ;,. 0 such that N(J, r);,:: re-J (recall that N(J, r) JS the least number of balls of radius r needed to cover the set J).
205
Dimension of Sets and Measures Invariant under Hyperbolic Systems Let 'R = { Rt. . .. , Rr} be a Markov partition for
f and U,. a Moran cover of
J by basic sets Rf.Jl = R;0 inc•;>, j = 1, ... , Nr. The sets CUl = x- 1 (R(il) = C; 0 inc"' > (where w; -= x- 1 (x;)) comprise a Moran cover of :E:A (recall that X is the c~ding map generated by the Markov partition). Repeating arguments in the proof of Theorem 13.1 one can show that there exist A > 0 and a positive integer N such that for any sufficiently small r, card {j : n(w;)
Consider an arbitrary cover
2::
g of I:~
= N};::: r 2•-ci.
by cylinder sets Cio
It follows that
'N
I:
j n(j)=N
where C 2 > 0 is a constant. We now have that for any n > 0 and N > n,
R(:E~,O,cp,Un,N)= I: C;o
I: C, 0
exp(
'NEg
sup
wEC;o
;NEQ
N
sup
iN
"f.
cp(a-"'(w)))
k=O
-
II iaukcx>>rd+Z• 2:: c2,
.,ERI'l k=O
where R(E1, 0, cp,Un., N) is defined by (A2.19') (see Appendix II) with a = 0 and cp(w) = -(d- 2e:) log la(x(w))l By Theorem 11.5 this implies that CP J ( -(d- 2e:) log Ia I) = PJ( -(d- 2e:) Jog Ia I) = PE~ ( -(d- 2e:) Jog Ia 0
xi) 2:: 0
and hence d- 2c ::; s. Since this inequality holds for all e: we conclude that d::; s. The last equality in Statement 1 follows from the variational principle (see Theorem A2.1 in Appendix II). This completes the proof of the first statement. We now prove the other two statements. Note that PJ( -slog Ia I)= hm(f)- s llog la(x)l dm(x) = 0,
where hm(f) is the measure-theoretic entropy ofm. One can use formulae (21.20) and {21.21) below to conclude that dm{x) = s form-almost every x E J The third statement follows now from Theorem 7.1. However, we present here a simple and straightforward proof of the third statement. Having in mind that the measure m is an equilibrium measure and
206
Chapter 7
PJ (-slog Jal) = 0, there exists a constant 0 3 any basic set ~ '· (x)
> 0 such that for any x
E M and
n
m(R.. '· (x)) $ 03
IT Ja(f"'xW"
(20.5)
k=O
(see Condition (A2.20) in Appendix II). Given r > 0, consider a Moran cover U,. = {RO)} of J constructed from a Markov partition R for f. It follows from the property of the Moran cover and (20.5) that m(B(x, r)) $
E
m(R!?l) $ M03 r 8 •
R(jlEU,
Thus, m satisfies the uniform mass distribution principle (see Section 7) and hence dimH m 2: s. The above arguments also imply the second statement. •
Remark. By slight modification of arguments in the proof of Theorem 20.1 one can strengthen the first statement of this theorem and prove the following result (see (Bar2]; compare to Statement 4 of Theorem 13.4): gwen an open set U C M
such that U n J f.
0
we have
dimH(U n J)
= dim8 (U n J) = dimn(U n J) = 8
(recall that 8 is the unique root of Bowen's equation PJ( -slog Jal) = 0). Analysing the proof of Theorem 20.1 one can also obtain a lower bound for the Hausdorff dimension as well as an upper bound of the upper box dimension of any set Z c J (which need not be invariant or compact). Namely,
where s and s are unique roots of Bowen's equations Pz (-slog Jai) = 0 and OPz( -slog JaJ) = 0 (existence and uniqueness of these roots are guaranteed by Theorem A2.5 in Appendix II). One can apply Theorem 20.1 to conformal expanding maps described in Examples 1-4 above. We leave it as an exercise to the reader to show that: 1) iff is a linear one-dimensional Markov map with the repeller J then
where s is the unique root of the equation Cl -•
(here c;
+ · · · +Cp-s
= 1
> 1 is the slope of f on the interval I;,
j = 1, ... , p).
Dimension of Sets and Measures Invariant under Hyperbohc Systems
207
2) iff is the induced map generated by conformal affine maps hl> . .. , hp: D -+ D (D is the unit ball in R"' ), h1(x) = >..ix + a1, then the number s (that is the common value of the Hausdorff dimension and lower and upper box dimensions) is the unique root of the equation
>.r" + ... + >.p' = 1. One can also use Theorem 20.1 to compute dimension of hyperbolic Julia sets of rational maps. We present two additional results (without proofs) that provide more detailed information on the Hausdorff dimension of Julia sets. The first result is due to Ruelle [R2] and deals with the two-parameter family of rational maps z >-+ zq- p Let Jq,p be the corresponding Julia set. Proposition 20.3. If Jq,p is hyperbolic then 2
dimH Jq P = 1 + - 1PI ' 4 1og q
+
(terms of order > 2 in p).
Consider a family {RA : >. E A} of rational maps. Given>., let J..\ be the corresponding Julia set The family R>. is said to be J-stable at >. 0 E A if there exists a continuous map h: A' x J-'o -+ C such that A' is a neighborhood of >. 0 in A and h(>., ·) is a conjugacy from (J>. 0 , R>. 0 ) to (J>., R>.) satisfying h(>.o, ·) = idiJ.x 0 • The second result was obtained by Shishikura [Shi] Consider the oneparameter family of complex quadratic polynomials R>.(z) = z2 + >.. The set 8M = {>.. E C : R..\ is not J -stable} is known to be the boundary of a set M c C called the Mandelbrot set. Proposition 20.4. There e:nsts a residual subset A C 8M such that dimH J >. = 2 for any >. E A. Estimates of Hausdorff Dimension and Box Dimension of Repellers: Non-conformal Case Let J be a repeller for an expanding Cl+"'-map f. If f is not conformal the Hausdorff dimension and the lower and upper box dimension of J may not coincide. An example is given by the induced map on the limit set of the selfsimilar geometric construction, described in Section 16.1 (or Section 16.3), which is smooth expanding but is not conformal. Nevertheless, a slight modification of the above approach, which is relied upon the thermodynamic formalism (and its non-additive version), can be still used to establish effective dimension estimates (i.e., estimates that can not be improved). We follow Barreira [Bar2]. Consider two HOlder continuous functions cp_ and cp on J
cp_(x) Let f_ and
= -log
lldxfll,
cp(x) =log ll(dxf)- 1 11.
t be the unique roots of Bowen's equations
(20.6)
Chapter 7
208
Let R = {R 1 , ... , Rp} be a Markov partition of J of a small diameter and f is a continuous expanding map we can consider two sequences of functions cp(k) and ~p(k) onE~ defined by (19.5). One can verify that given e > 0, there exi;i;s k 2:: 0 such that for every w E E~ and n ;::: 1 we have (E~, u) the symbolic representation of J by a subshift of finite type. Since
-ne +
n
n
j=O
j=O
L 1£.(/i(x)) ~ r£.kl(w) ~ -q;~kl(w) ~ ne + LW(Ji(x)),
where x ~ x(w) E J. This implies that t ~ flkJ ~ s 1 such that
for any x E J and n 2:: 1, where K = max{l[d.,/11: x E J}. We wish to apply the non-additive version of thermodynamic formalism and find roots of Bowen's equations (20.9) In order to do this we ought to establish Property (12.1). Given 15 E (0, 1], we call the map f 15-bu.nched if for every x E J we have (20.10)
Proposition 20.5. [Bar2] Iff is a Cl+"'-expanding 15-bunched map (for some 8 > 0} then the sequences of functions cp and cp satisfy Condition (12.1}. Moreover, there exists e ;::: 0 such that for e;ry w E E~,
for all sufficiently large n and some k ;::: 1 {where x = x(w)).
In view of Proposition 19.1 this imphes dimension estimates for the repeller J, (20.11)
Dnnension of Sets and Measures Invariant under Hyperbolic Systems
209
where§. and 8 are unique roots of Bowen's equations {20.9) (one can show that under the assumptions of Proposition 20.5 Bowen's equations (20.9) have unique roots). The lower and upper estimates in (20.7) and (20.11) cannot be improved. Note that n n
L~(Ji(x)) ~ ~(x) ~ "fPn(x) ~ L"fP(Ji(x)) i=O
j=O
for every x = x(w) E J and n ;::: 1. These inequalities rmply that if I is an 15-bunched Cl+"-expanding map then t ~ §. ~ s ~ t. If f is conformal it is easy-to see that t = §. = s = t. If f is not conformal the numbers §. and s may provide sharper estimates then the numbers~ and t. Indeed, in (Bar2], Barreira constructed an example of a !-bunched C 00 -expanding map of a compact manifold for which t < §. = 8 < t. This map is not conformal but can be shown to be weakly-conformal (as a continuous expanding map; see (19.7)). 21. Multifractal Analysis of Gibbs Measures for Smooth Conformal Expanding Maps We undertake the complete multifractal analysis of Gibbs measures for smooth conformal expanding maps. Let J be a conformal repeller for a Cl+_ conformal expanding map I· M ~ M of a compact smooth Riemannian manifold M. We assume that f is topologically mixing. The general case can be reduced to this one (with obvious modifications) using the Spectral Decomposition Theorem.
Thermodynamic Description of the Dimension Spectrum Let 'R = {Rl> ... , Rp} be a Markov partition for f and X the corresponding coding map from E1 to J (see Section 20). Consider a Holder continuous function cp on J. The pull back by x of cp is a Holder continuous function rp on E1, i.e., rp(w) = cp(x(w)) for wE Et Since f is topologically mixing (and so is the shift u) an equilibrium measure corresponding to this function, 11- = J-L.p, is the unique Gibbs measure for u (see Appendix II). Its push forward is a measure on J which is a unique equilibrium measure corresponding to cp. We denote it by 11 = 11v>. Define the function '1/J on J such that log '1/J = cp - PJ (cp) Clearly '1/J is a Holder contrnuous function such that PJ(Iog '1/J) = 0 and 11 is a unique equilibrium measure for log'I/J. By the variational principle (see Theorem A2.1 in Appendix II) we obtain that
1log'I/J(x)diJ.(x) = h (f) 11
= h,_.(u).
Define the one parameter family of functions cpq, q E ( -oo, oo) on J by
cpq(x) = -T(q) log ia(x)i + q log'I/J(x), where T(q) is chosen in such a way that PJ(cpq) = 0. It is obvious that the functions cpq are Holder continuous. Clearly, T(O) = dimH J = s.
Chapter 7
210
The function T(q) can also be described in terms of symbolic representation of the repeller by a subshift of finite type (E1,u). Namely, let i:i, ~~and cpq be the pull back by the coding map x of the functions a, 'if;, and )
q
'
where the partial derivatives are evaluated at (q, r) = (q, T(q)). We use the explicit formula for the second derivative of pressure for the shift map on E~ obtained by Ruelle in [Rl]. Namely,
Chapter 7
212 where Qh is the bilinear form defined for ht, h2 E G" (E,;t, R) by
and /1-h is the Gibbs measure for the potential h. Ruelle also showed that Qh(g,g) 2: 0 for all functions g E G"'(E1,JR) and that Qh(g,g) > 0 if and only if g is not cohomologous to a constant function (see definition of cohomologous functions in Appendix V). Applying the second derivative formula we obtain that -
()2
82 r P(-(ro +et +c2)logliil +qlog'ljl), 8 2 = Qh(log liil, log liil).
or 2 P(-rlogliil +qlog'ljl) =
Arguing similarly we find that 82 Bq 2 P(-rlogliil + qlog'ljl) = Qh(log'ljl, log¢),
82 q rP(-rlog liil 8 8
-
-
+ qlog'ljl) = Qh(log'ljl,
logliil).
This implies that T"(q)
= Qq(logti}- T'(q) log Iii!,
logti}- T'(q) log !iii)
JE+ log liild~tq A
It follows that T"(q) 2: 0 for any q Moreover, T"(q) > 0 for some q if the function logti}(w) -T'(q) log lii(w)! is not cohomologous to a constant function. The latter can be assured provided that the functions log'¢(w) and - T' (q) log Iii(w) I are not cohomologous. On the other hand, if they are cohomologous for some q then
Hence, -T'(q)
= s.
This implies that v
= m.
•
Diametrical Regularity of Equilibrium Measures An important ingredient of the multifractal analysis of equilibrium measures is the remarkable fact that these measures are diametrically regular (see Condition (8.5)) as the following statement shows.
Proposition 21.4. Let cp be a Holder continuous function on a conformal repeller J. Then any equilibrium measure for tp with r-espect to f is diametrically regular.
Proof. Let v = v'f' be an equilibrium measure for cp. Choose a Markov partition R of J. Given a number r > 0, consider a Moran cover .U,. of J. Fix a point x E J
213
Dimension of Sets and Measures Invariant under Hyperbolic Systems
and choose those elements R(l), ... , R(m) from the Moran cover that intersect the ball B(x, 2r). We recall the following properties of the Moran cover: for every j = l, ... ,m, (1) R(i) = Ri, in :(x), where q is chosen such that a= a(q). The set j - the irregular part of the multifractal decomposition consists of points with no pointwise dimension. We will see in Appendix IV that j =f:. 0; moreover, it is everywhere dense in J and carries full Hausdorff dimension (i.e., dimH j = dimH J) and full topological entropy (i.e., hJ(f)hJ(/)).
Dimension of Sets and Measures Invariant under Hyperbolic Systems
223
Pointwise Dimension of Measures on Conformal Repellers Given a point x E J, we define the Lyapunov exponent at x by
>.(x)= lim logJJd/~11 >0 n-+oo n
(21.19)
(provided the limit exists, see Section 26 for more details). If vis an !-invariant measure then by the Birkhoff ergodic theorem, the above limit exists v-almost everywhere, and if v is ergodic then it is constant almost everywhere We denote the corresponding value by Av > 0. Let v be an equilibrium measure corresponding to a Holder continuous function on J. It follows from Statement 1 of Theorem 21.1 that for v-almost every X E
J, - h,(f) d v (X ) '
(21.20)
>..,
where h,(J) is the measure-theoretic entropy off and
>.,
=
1 ElogJa(Jk(x))J n limn-toon k=O
=
1
logJa(x)Jdv(x).
(21.21)
J
F\1rthermore, let G., be the set of all forward generic points of v (i.e., points for which the Birkhoff ergodic theorem holds for any continuous function on X; see Appendix II). It follows from Lemmas 2 and 3 in the proof of Theorem 21.1 (applied to the measure v 1 = v) that for every x E G,,
In view of Theorems 7.1 and 7.2, this unpiles that
hv(J) = d"imHV = d"imH G v·
~
(21.22)
We extend (21.20) and (21.22) to any Borel ergodic measure on J of positive measure-theoretic entropy which is not necessarily an equilibrium measure.
Theorem 21.3. For any Borel ergodic measure v of positive measure-theoretic entropy supported on a conformal repeller J we have (I) the equality {21.20} holds for almost every x E J; (2) h';.~f) = dimH v = dimH G, = dim 8 G, = dimsG,... Proof. Set d,.. = h-;.SD. We first show that d,(x) ;::: d,.. The proof is a slight modification of the proof of Statement 2 of Theorem 13.1. Consider a Markov partition = { R" ... ' Rp} for I and the corresponding symbolic model (E1, u) (see Section 20). Let 11- be the pullback to E~ of the measure v by the coding map X
n
Chapter 7
224
Fix c > 0. It follows from the Shannon-McMillan-Breiman theorem that for JL-almost every wE E_i one can find N1(w) > 0 such that for any n ~ N 1 (w),
JL(C; 0
;,.
{w)) ::; exp( -(h- c)n),
{21.23)
where Cio ;,.(w) is the cylinder set containing wand h = h,.(O") is the measuretheoretic entropy of the shift u. It follows from the Birkhoff ergodic theorem, applied to the function log la(x)l, that for v-almost every x EM there exiSts N 2 (x) such that for any n ~ N 2 (x),
1 M
n
1 n log la(x)ldv::; -log la(Ji(x))l n i=l
+€
(21.24)
In order to prove the desired lower bound for d.,(x) it remains only to use (21.23) and (21.24) and to repeat readily the argument in the proof of Statement 2 of Theorem 13.1. We now prove the opposite inequality. Fix 0 < r < 1 By (20.1) it follows that R, 0 ;,. 0 is a constant Therefore,
where C2 > 0 is a constant. By virtue of (20.1) we obtain for all x E J,
-d ( ) _ 1. logv(B(x, r)) < d liD l _ v 11 X r-+0 ogr
+ 2c.
Since c: can be arbitrarily small this proves that 4... (x) ::; d., In order to prove the second statement we note that dv = dimH v ::; dimH 0 11 • On the other hand, by (A2.16) (see Appendix II), we conclude that 0 =h.,(!)JJ d...>...,dv = Po"(-dv>...,), i.e., d., is the root of Bowen's equation. Repeating arguments in the proof of Theorem 20.1 (applied to the set Gv instead of J) we obtain that dimBGv ::; d.,. This completes the proof of the theorem. • Some results, similar to Theorem 21.3 for measures supported on conformal repellers of holomorphic maps, were obtained in [Ma2, PUZ]. Since the measure-theoretic entropy is a semi-continuous function we obtain as an immediate corollary of Theorem 21.3 and (21.21) that. the Hausdorff dimension of a Borel ergodic measure v on a conformal repeller is a semicontinuous function of 11.
Information Dimension We compute the information dimension of a Gibbs measure 11 on a conformal repeller J. Applying Theorem 21.1 and taking into account that the function T(q) is differentiable we obtain that the limit lim T(q) 1- q
q-+1
Dimension of Sets and Measures Invariant under Hyperbolic Systems
225
exists and is equal to -T'(l) = a:(l). As we know the latter coincides with the Hausdorff dimension of v (see (21.22) and Remark 3 above) This implies that
fv(a:(l))
= a:(l) = -T'(l) = I(v) = l(v),
where l(v) and 1(v) are the lower and upper information dimensions of v (see Section 18). We note that Statement 5 of Theorem 21.1 allows one to extend the Hentschel-Procaccia spectrum and Renyi spectrum for dimensions for any q f= 1 Moreover, the above argument makes it possible to define these spectra even for q = 1 (as being equal to a:(l)). Dimension Spectrum for Lyapunov Exponents We consider the multifractal decomposition of the repeller J associated with the Lyapunov exponent >.(x) (see (21.19)
(21.25)
where
L = {x E J: the limit in
(21.19) does not exist}
is the irregular part and LfJ = {x E J: the limit in (21.19) exists and >.(x) =
{3}.
If v is an ergodic measure for f we obtain that >.(x) = >., for v-almost every x E J. Thus, the set L>." f= 0. Moreover, if v is an equilibrium measure corresponding to a Holder continuous function the set L>.u is everywhere dense (si.nce in this case the support of v is the set J). We note that if the set LfJ is not empty then it supports an ergodic measure VfJ for which .>.,~ = {3 (indeed, for every x E Lp the sequence of measures ~;;;;~ &J•(x) has an accumulation measure whose ergodic components satisfy the above property). There are several fundamental questions related to the above decomposition, for example, (1) Are there points x for which the limit in (21.19} does not exist, i.e., Lf= 0? (2) How large is the range of values of .>.(x)? (3) Is there any number {3 for which any ergodic meMure v with v(L{3) > 0 is not an equilibnum measure?
k
In order to characterize the above multifractal decomposition quantitatively, we introduce the dimension spectrum for Lyapunov exponents of f by
£(.8) = dimn Lp.
Chapter 7
226
This definition is inspired by the work of Eckmann and Procaccia [EP). In [We), Weiss derived the complete study of the Lyapunov dJmension spectrum for conformal expanding maps by establishing its relation to the f vmo.x (a )-spectrum, where Vma.x is the measure of maximal entropy. Notice that the measure of maximal entropy is a unique equilibrium measure corresponding to the function cp = 0, and hence t/J =constant= exp(-hJ(J)), where h;(f) is the topological entropy of f on J. Therefore, for every x E L13,
This implies the following result.
Theorem 21.4. [We) Let J be a conformal repeller for a smooth expanding map f. Then (1) If Vmax f. m (m is the measure of full dimension) then the dimension spectrum for Lyapunov exponents
£({3) =
fvmax
CJ~f))
is a real analytic strtctly convex function on an open interval (fh, f32] containing the point f3 = hJ(f)fs. (2) If Vmax = m then the dimension spectrum for Lyapunov exponents is a delta function, i.e.,
l(f3)
= { s, 0,
for f3 for f3
= h;(f)/ s
f.
h;(f)/s
As immediate consequences of Theorem 21.4 we obtain that if Vmax f. m then the range of the function A.(x) contains an open interval, and hence the Lyapunov exponent attains uncountably many distinct values On the contrary, if the Lyapunov exponent A.(x) attains only count ably many values, then Vmax = m. There is an interesting application of this result to rational maps. In [Z), Zdunik proved that in the case of rational maps the coincidence llmax = m implies that the map must be of the form z -+ z±n. Therefore, we obtain the following rigidity theorem for rational maps.
Theorem 21.5. If the Lyapunov exponent of a rational map with a hyperbolic Julia set attains only countably many values, then the map must be of the form z-+ z±n. We can now answer the above questions. Namely, (1) the set L is not empty and has full Hausdorff dimension (see Appendu IV; compare to {21.18}); (2) the range of values of A.(x) is an interval ({31> f3z] and for any f3 outside this interval the set L13 is empty (i.e., the spectrum is complete); (3) for any {3 E (f3t, {32} there exists an equuibnum measure v corresponding to a Holder continuous function for which v(L(J) = 1.
Dimension of Sets and Measures Invariant under Hyperbolic Systems
227
22. Hausdorff Dimension and Box Dimension of Basic Sets for Axiom A Diffeomorphisms
Axiom A Diffeomorphisms In this section we study the Hausdorff dimension and box dimension of sets invariant under smooth dynamical sy:;tems with strong hyperbolic behavior. Let M be a smooth finite-dimensional Riemannian manifold and f: M --+ M a Cl+"'-diffeomorphism (i.e., f is a Cl+"-invertible map whose inverse is of class Cl+"). A compact /-invariant set A C M is said to be hyperbolic if there exist a continuous splitting of the tangent bundle TAM = E(•) ® E(") and constants C > 0 and 0 < ). < 1 such that for every x E A (1) dfE(•l(x) = E(•l(J(x)), dfE(ul(x) = E('-'l(J(x)); (2) for all n
~
0
lldrvll S C).nllvll lidf-nvll S C).nllvJI
if v E E(x). if v E E("l(x).
The subspaces E(•l(x) and E(x)
= U rn(W1~~(fn(x))),
wCu>(x)
= U r(W1~"j(f-n(x))).
n~O
n~O
They can be characterized as follows:
as n -too},
w<sl(x) ={yEA: p(r(y),r(x)) -t 0
wCul(x) ={yEA: p(f-n(y),rn(x)) -t 0
as n-+ oo}.
(22.4)
A diffeomorphism f is called an Axiom A diffeomorphism if its nonwandering set n(f) is a locally maximal hyperbolic set (see Section 20 for definition of a non-wandering set). The Spectral Decomposition Theorem claims (see [KH]) that the set n(f) can be decomposed into finitely many disjoint closed /-invariant locally maximal hyperbolic sets, fl(f) = A1 U · · · U Am, such that f I A; is topologically transitive. Moreover, for each i there exiSt a number n; and a set A; C A; such that the sets Jk(A;) are disjoint for 0 s k < n;, their union is the set A;, r•(A;) =A;, and the map r· I A; is topologically mixing. Let A be a locally maximal hyperbolic set of f. From now on we assume that f I A is topologically mixing. This assumption is technical and we require it to simplify arguments iJ?. the proofs The general case can be reduced to this one by using the Spectral Decomposition Theorem. . A non-empty closed set RcA is called a rectangle if diamR s 8 (where 8 is chosen as in (22.3)), R = intR, and [x, y] E R whenever x, y E R. By Property {8) a rectangle R has "direct product structure", i.e., given x E R, there exists a Holder continuous homeomorphism {22.5)
We remind the reader that a finite cover R Markov partition for f if (1) intR. n intR; = 0 unless i = j; (2) for each x E intR; n f- 1 (intR;) we have
= {R 1 , ••. , Rp}
f(W1~1(x) n R;) c W1~2(!(x)) n R;, f(W1~"J(x) n R;) ::> W1~':/(f(x)) n R;.
of A is called a
Dimension of Sets and Measures Invariant under Hyperbolic Systems
229
n
A Markov partition = {RI> ... ,Rp} generates a symbolic model of A by a subshift of finite type (EA, u), where EA is the set of two-sided infinite sequences of integers which are admissible with respect to the transfer matrix of the Markov partition A= (a.;) (i.e., a;1 = 1 if intR; n r 1(intR;) ::/= 0 and a,; = 0 otherwise; see Appendix II). Namely, define
Now we define the coding map
x: EA -t A by
x( ... i_n·· io .. i,. ... )=
nR.-n
in•
n 0. For almost every y E R(x) the conditional measure v(y) generated by 11 on wi;l(y) nR(x) (respectively, the conditional measure v(x) containsaballinW1~:>(x) ojradiusr..;0 i. and is contained in a ball in W1~"j (X) of radius 1';0 in ; (2) There exist positive constants K1 and K2 such that for every basic set (u) d n = x(Cj">),j = 1, ... , Nr may overlap along their boundaries and comprise a cover of A C3(u) = C; 0
;
< ·l. ~
These sets form a disjoint cover of
if it does not cause any confusion). We have that
R]">
~
=
R;0
'";l
for some
Yj E A(u)(x).
Let Q C C"!" be a subset (not necessarily invariant). One can repeat the above arguments"' to construct a Moran cover of the set Q. It consists of cylinder sets cJ"> ,j = 1, ... ,Nr for which there exist points Wj E Q such that cj"> =
Cio ;,. and the intersection
cj"> n cfu.> n Q
is empty as soon as i =f. j. We
denote this cover by~~~· Moran covers have the following crucial property Given a point z E A(u) (x) and a sufficiently small r > 0, the number of basic sets R]"> in a Moran cover that have non-empty intersection with the ball B(u)(z, r) is bounded from above by a number M which is independent of z and r. We call this number a Moran multiplicity factor. In order to establish this property let f = max {diam R. : i = 1, ... , p}. Since the sets R. are the closure of their interiors there eXIsts a number 0 < r 1 ~ f such that for every x E A,
m">
The desired property of Moran covers follows from Proposition 22.3.
Hausdorff Dimension and Box Dimension of Locally Maximal Hyperbolic Sets for u-Conformal and s-Conformal Diffeomorphisms Let A be a locally maximal hyperbohc set for a C 1+a.diffeomorphism f. Assume that f is topologically mixing. We denote by ,.(u) a unique equilibrium measure corresponding to the Holder continuous function -t(u) log Ja(u)(x)J on A, where t;
Dimension of Sets and Measures Invariant under Hyperbolic Systems
233
(2) t(u.) _
h,.(u)
(f)
- J,.. log iaukcxJ)I-d+2e 2:: c2,
G; 0 , ;NE9 o:ER~;J k=O
where R(Cf, 0, cp,UN, N) is defined by (A2.19') (see Appendix II) with a = 0 •o and cp(w) = -(d- 2c) log JaJ) 2::
0
and hence d- 2c ::; t(x)(R~:!.i. nwomorphism f. Assume that f is s-conformal. Denote by , a unique equilibrium measure corresponding to the Holder continuous function t + t. > 1 is a constant. One can show that the set A=
n
r(D.')
nEZ
is a locally maximal hyperbolic set with "almost" vertical unstable and "almost" honzontal stable subspaces (see [KH]). In the two-dimensional case the above construction was first introduced by Smale [Sm] and the set A is known as a "Smale horseshoe " We describe the topological structure of A. Note that for every n 2': 0 the set fn(l:!.') consists of pn simply connected close disjoint "almost" vertical components which we denote by 1:1.~:) in. Similarly, for every n 2': 0 the set J-n(l:!.') consists of pn simply connected close disjoint "almost" horizontal components which we denote by !:!.~~ in (see Figure 18). Given a point z E A, denote by 1:1.!:) in (z) and !:!.l:) ;. (z) the vertical and, respectively, horizontal component that contains z (clearly, it is uruquely defined). One can show that (1) for every z E A the set nn>O 1:1.~:)
in (z)
is a smooth £-dimensional unstable
submanifold that is isom~phic to B2; similarly, the set nn;::o !:!.~) ;n (z) is a smooth k-dimensional stable submanifold that is isomorphic to B 1 ; (2) every point z E A can be coded by a two-sided infinite sequence of integers ( ... L 1 ioi 1 ..• ), ii = 1, . .. ,p such that
moreover, the coding map X: A--+ Ep defined by x(x) = ( .. L1ioh .. is bijective and onto. The map x establishes the symbolic representation of the horseshoe by the full shtft on p symbols. There is another description of the horseshoe which is more smtable for studying its dimension. Notice first that the sets
define a geometric construction in 1:!. (modeled by the full shift on 2p symbols) whose limit set is A. Furthermore, consider the sets
They define geometric constructions in B 1 and B 2 respectively (modeled by the full shift on p symbols). We denote these constructions by ca(y)
= 11'I(/- 1 (11'1 1 (y) n D.;)),
g~u>(x)
= 11'2(r 1 (11'2 1 (x) n D.;)).
(23.1)
On the other hand, let CG(•) and CG(u) be two geometric constructions in B1 and B 2 modeled by the full shift on p symbols and generated by affine maps g1•l and g}u) respectively. Let also p(•) and p(u) be their limit sets. One can build a linear horseshoe A for a map f such that A= p x p(u) and relations (23.1) hold. If the maps ui•l and g~u) arc conformal then by Theorem 22.2, we obtain that dimH A = dim 8 A =dims A= dimH p(•) + dimH p(u) . On the other hand, using Example 16.1 we conclude that there exist a linear horseshoe in JR4 for which dimH A < dim 8 A = dimaA . We also remark that if the ratio coefficients of the affine maps g~•) and g~u) are equal then the coding map is an isometry between the the horseshoe A and l::P. Thus, it preserves the Hausdorff dimension and box dimension (compare to Theorem A2.9 in Appendix II). Three-Dimensional Solenoids
We follow Bothe [Bot). Let 8 1 be the unit circle and 7) 2 the unit disk in Then v = 8 1 X 'D 2 is a solid torus. The projections 11' : v -+ 8 1 and R 1 Pl, P2 : V -+ 8 X (-1, 1] are defined by 2•
1r(t,x,y)=t,
Pl(t,x,y)=(t,x),
P2(t,x,y=(t,y).
Dimension of Sets and Measures Invariant under Hyperbolic Systems
241
We also denote by V(t) = {t} x 'D 2 = u- 1 (t) (where t E S 1 ). Let '9 be the space of all 0 1-embeddings f : V ~ V of the form
j(t,x, y) = (cp(t), At(t)x + zt(t), .X2 (t)y + z2(t)),
(23 2)
where
are C 1-maps, and
cp IS expanding, i.e. dcp > 1.
dt The last condition implies that the degree (} of
(w+)
=-
lim log p.(C;, in), p.{ C;o in)
n-+oo
where w+ = (ioil ... in ... ) E E1. According to Appendix II the above limits exist and the functions log~ and logr,b
±oo
We observe that since the spectrum Vv is not a delta function, Tv(q) is not linear and is strictly convex. Hence, a+ < dimH J < tL. Therefore, raising both sides of the equation (A4.3) to the power 1/q and letting q-+ ±oo we obtain max{/31 c1 "'+, f32c2 "'+} = min {/31 c1 "'-, f32c2 "'- } = 1. We ll8sume that f3tct''+ = 1 (the case /32 c2 "'+ = 1 can be treated in a similar way; in this case, pis the involution automorphism). Since a+ < a_ we must have /32c2"'- = 1. Setting q = 0 and q = 1 in the equation (A4.3) we obtain respectively, Ct- dimH J
+ c2- dimn J = 1
Set x = c1- dimn 1 , 'Y =a+/ dimH J can easily derive the equation x"~
and
f3t
+ /32
= 1.
< 1, and b = a_f dimH J > 1. Then one
+ (1- x)b = 1.
We leave it ll8 an easy exercise to the reader to show that this equation has a unique solution x E (0, 1) which uniquely determines the numbers c 1 and c2 and hence also the numbers f3t and /32. •
269
A General Concept of Multifractal Spectra
Remarks. (1) We have shown that for a one-dimensional linear Markov map of the unit interval one can determine the four numbers /31> /32, c1, and c2 using the spectrum T>v. If instead the spectrum CE is used then only the numbers fJ1 and !32 can be recovered: one can show that if /31 ~ 132, then /31 = exp lim (TE(p)/p) p--t+oo
and
lim (TE(P)/p). 132 = exp p-+-oo
In a similar way, using one of the spectra Cv or CE one can determine only the numbers c1 and c2. (2) The multifractal rigidity for two-dimensional horseshoes is demonstrated in [BPS3].
Chapter 8
Relations between Dimension, Entropy, and Lyapunov Exponents
In the previous chapters of the book we have seen that the pointwise dimension is a useful tool in computing the Hausdorff dimension and box dimension of measures and sets. The key idea is to establish whether a measure is exact dimensional (i.e., its lower and upper pointwise dimensions coincide and are constant almost everywhere). If this is the case then by Theorem 7.1, the Hausdorff dimension and lower and upper box dimensions of the measure coincide. This also gives an effective lower bound of the Hausdorff dimension of the set which supports the measure. One can obtain an effective upper bound of the Hausdorff dimension of the set by estimating the lower pointwise dimension at every point of the set and applying Theorem 7.2. In the previous chapters of the book we described several classes of measures, invariant under dynamical systems, which are exact dimensional. We also obtained formulae for their pointwise dimension. These classes include Gibbs measures concentrated on limit sets of geometric constructions (CPW1-CPW4) (see Theorem 15.4), invariant ergodic measures supported on conformal repellcrs (see Theorem 21.3) or two-dimensional locally maximal hyperbolic sets (see Theorem 24.2). In [ER], Eckmann and Ruelle discussed dimension of hyperbolic measures (i.e , measures invariant under diffeomorphisms with non-zero Lyapunov exponents almost everywhere). This led to the problem of whether a hyperbolic ergodic measure is exact dimensional. This problem has later become known as the Eckmann-Ruelle conjecture and has been acknowledged as one of the main problems in the interface of dimension theory and dynamical systems. Its role in the dimension theory of dynamical systems is similar to the role of the Shannon-McMillan-Breiman theorem in ergodic theory. In [Y2], Young showed that hyperbolic measures invariant under surface diffeomorphisms are exact dimensional. Later Ledrappier [L] proved theis result for glc'neral Sinai-Ruelle-Bowen measures In [PYJ, Pesm and Yue extended his approach to hyperbolic measures satisfying the so-called semi-local product structure (see below; this class includes, for example, Gibbs measures on locally maximal hyperbolic sets). Barreira, Pesin, and Schmeling (BPSl] obtained the complete affirmative solution of the Eckmann-Ruelle conjecture which we present in this chapter. We also demonstrate that neither of the assumptions in the Eckmann-Ruelle conjecture can be omitted. Ledrappier and Misiurewicz (LM] constructed an ex270
Relations between Dimension, Entropy, and Lyapunov Exponents
271
ample of a smooth map of a circle preserving an ergodic measure with zero Lyapunov exponent which is not exact dimensional (see Example 25.3 below). In [PWl], Pesin and Weiss presented an example of a Holder homeomorphism whose measure of maximal entropy is not exact dimensional (see Example 25.1 below). Barreira and Schmeling [BS] showed that for "almost" any Gibbs measure on a two-dimensional hyperbolic set the set of points, where the lower and upper pointwise dimensions do not coincide, has full Hausdorff dimension (see Appendix IV). 25. Existence and Non-existence of Pointwise Dimension for Invariant Measures We begin with examples that illustrate some problems of the existence of pointwise dimension. Our first example shows that the pointwise dimension may not exist even for measures of maximal entropy invariant under a continuous map. Example 25.1. [PWI] There exists a geometnc construction with rectangles modeled by the full shift on two symbols (see Section 16} such that the corresponding induced map G on the limit set F of the construction is a Holder continuous endomorphism for which the unique measure of mllXImal entropy m satisfies 4m(x) < dm(x) for almost every x E F. Proof. We begin with the geometric construction presented in Example 16.2. Note that for this construction de£ S
-
=
S>,
-
log2 -logd
= ---1
log2 s = sx= ----· -log.\
_de£
Hence, the functions 0 and v(B) > 0. One can check that for every x E A,
E1.
!
lim log l!!..n(x)l :::; lo ('Yf3) n-+oo n 2 g and for every x E B,
r
n~~
loglt!.n(x)l
n
11
( o)
= 2 og a
·
Since x- 1 (6.n(x)) is a cylinder set the Shannon-McMillan-Breiman theorem implies that for v-abnost every x E F,
_ lim log v(.!ln(x)) = h,_.(e1) > O. n-tOO n Thus, by Theorem 15.3 h (u) 2h,_.(CT) for abnost every x E A d(x) =d(x) = -liffin-+oo(logl~n!z!) = -log(f3'Y) and d( ) _ d( ) _
- x -
x --lim
h,.(u) _ 2h,.(C1) (Iogl6.ft(xl1) - -log(ao)
n-+oo
for abnost every x E B.
n
The first statement follows now from Theorem 15.3. Repeating the arguments in Section 25.1 one can define the HOlder homeomorphism G and show that it possesses an invariant Borel ergodic measure i/ with respect to which d;;(x, y) = 4;;(x, y) d;;(x, y) for i/-almost every (x, y); however, the function d;;(x, y) is not essentially constant. This implies the second statement and completes the construction of the example. •
=
Relations between Dimension, Entropy, and Lyapunov Exponents
275
Remark. The geometric construction described in Example 25.2 is called an asymptotic Moran-like geometric construction since its ratio coefficients admit asymptotic behavior (25.1) It is proved in (PWl] that S>..::; dimn F, where S>. is a unique root of Bowen's equation PE+(slog>.;,) = 0. Example 25.2 illustrates A that the strict inequality can occur. Indeed, one can easily check, using Statement 1 of Theorem 13.3, that s>.. = Jog 2/log(afry6). On the other hand, one can compute, using Theorem 15.2, that . { log 2 log 2 } dtmn F = max -log(}3-y), -log(a6) . The following example demonstrates non-existence of pointwise dimension for smooth maps. It was constructed by Ledrappier and Misiurewicz in [LM]. Example 25.3. For every integer r ~ 1 there exzst a C'" -map f of the interval [0, 1] and f-invanant ergodic measure 1-1. for which !t,.(x) < d11 (x) for almost all points x. Proof. We outline the proof in the case r = 1. Choose numbers A, B, and C such that A > 1 and 0 < B < C < A!t. Let On be a sequence of numbers satisfying 81 = 1 and B::; On+l/On ::; C. Since C < 1 the sequence On decreases towards 0. We define two sequences of points {an}, {bn} of the interval (0, 1] 0 = az
6,.+1 and thus, 1 = 'Yt > 61 > 62 > · · · -+ 0. Therefore, we can define f as a linear map on each interval L,. with the slope >.,. given as
Note that
ro, 11 = {c} u
(uL. ) (u u
n;:=:l
M .. )
n?;l
and it remains to define f on the intervals M,.. Denote
For n even we set
f (b,. + ~IMnl)
f'
= f(b,.) + ~IM,.i(>.,. +a,.),
(b.,+ ~IMnl) =a,..
One can now define f separately on [b,., b,. + ~IM,.I) and [b,. + ~IM,.I, a,.+2) to obtain a C00 -function on [b,., a,.+2] (see details in [Mi]). For n odd one can define f analogously. This gives a map f on (0,1]. The condition on f to be of classC 1 is We have !Ani= A2-n-+ 0 as n-+ oo. Furthermore,
Thus,
lwnl-+ 0 as n-+ oo and hence
as n-+ oo. Let K,. be the interval [a,.,a,.+ 1] or (an+t 1 a,.]. We need the following properties of sets K,. (see {Mi]).
Relations between Dimension, Entropy, and Lyapunov Exponents
277
Lemma 1. (1) The sequence of sets (Kn)~=l is decreasing; (2) f 2 n-l (Kn) = Kn, f 2n-l (Kn+d = L,.; (3) The sets J'(K,.), i = 1, ... , 2n-l are d'tBjoint; (4) p"-'+'(Kn+I) n J'(Kn+I) = 0 fori= I, ... , zn- 1 ; (5) The map j'- 1 1/(K,.) is linear fori= 1, ... , 2"- 1 . For each i
n-1
= 1, ... , 2n-l we write i = 1 + L: ek(i)2k-l
with ek(i)
= 0 or 1.
k=j
Let Lemma 2. For all n
~
1 and i = 1, ... , 2n-l we have n-1
lfi(Kn)l
= IT (k(ek(i)). k=l
Proof of the lemma. We use induction on n. For n = 1 the result is obvious. Assume that the statement holds for n = m. We shall prove it for n = m + 1. Observe that for i = 1, ... , 2m-l the set /'(Km) is a disjoint union (modulo endpoints) of sets J'(Km+l), t•(Lm) = / 2m-'+i(Km+I), and a remaining gap
Gi,m·
.
Observe also that the map /'- 1 is linear on /(Km) so that the lengths of these intervals are in the same proportions as
1/(Km+l)l IJ(Km)l
=
'Ym+l I'm
1 llm+l
IJ(Lm)l
=A 0::: ' lf(Km)l =
I'm -Om I'm
The induction follows clearly from the above two observations.
Om+l
= 9:;: . •
This lemma gives us information on the lengths of intervals building the at tractor for the map j. The following lemma provides estimates of the lengths of the gaps Gi,m· Denote
Clearly, /3 > 0.
Lemma 3. For all m
~
2 and i
= 1, ... , 2m-l
we have
Proof of the lemma. Note that by linearity of J'- 1 1/(Km) the above inequalities are equivalent to
Chapter 8
278 or
< {3 [1 - ( 1 + ~) Om+l] . Om A Om ity follows from the inequal second The The first inequality holds since A > 1.
~ Om+l
.(x), and k; (x), i = 1, ... , q are measurable and invariant under f. Let p be a Borel /-invariant measure on M. We will always assume that p is ergodic Then the function q(x) = q is constant JL-almost everywhere and so are the functions >.(x) and k;(x), i = 1, . .. ,q. We denote the corresponding values by ,\~) and k~). A measure J1. is said to be hyperbolic if
Chapter 8
280
for some k, 1 ::; k < q. If J1- is such a measure then for Jl--almost every point x EM there exist stable and unstable subspaces E(x), E("l(x) C T.,M such that
(1) E(x)
=
E(u) (J(x) );
(2) for any n 2': 0, ildf~vii
::; Cl(xhnllvll ildf;nvll::; CI(xhnllvll
if v E E(x), ifv E E(x)) 2': C 2 {x) > 0, where C2 (x) is a measurable function and L denotes the angle between subspaces E(x) and E(x,rt) centered at x of radius rein the intrinsic topology generated by the Riemannian metric;
(8) there exists a function £1 = '1/J(l) < oo, such that for any x, y E Ae, if the intersection B(x,rt) n B("l(y,re) f. 0, then it consists of a single point
z E Ae,; (9) there exists K
> 0 such that for any
x E A1 ,
p(x, rt) n B("l(y, re) is not empty and consists of a single point z E II (one can see that II c At,). Given a rectangle II and points x, y E II, define the u-holonomy map s(x, rt) n II -+ s(y, rt) n II by H~~J(z) = B(x) = II n B 0 and 1-L-almost every x, y E II We fix such a rectangle II. According to Proposition 26.2 it is sufficient to show that 4(1-') 2:: d(s) + d(u).
Relations between Dimension, Entropy, and Lyapunov Exponents
283
Proposition 26.2 (that states the existence of the limits in {26.2)) implies that there exist a closed set A 1 c II with !k(A1 ) > 0 and a number r 1 > 0 satisfying the following condition: (10) for any 0 < r ~ r1 and any x E A1, (26.3)
It follows from the Borel Density Lemma (see Appendix V) that one can find a closed set A2 c A1 with tt(A2) > 0 and r2 > 0 such that for any 0 < r :0:::: r 2 and any x E A2, tt(B(x, r)) ~ 2p,(B(x, r) n A 1 ). (26.4) Fix a point Xo E A2 for which J.l.~~) (WI~") (xo)
n A2) > 0 and
lim logJ.!.(B(xo,r)) = d(J.I.). log r -
;:::;a
We first study the factor measure ji. induced by the measurable partition e-e;J(B(ul(xo, Kr)} :::; (2Kr}d-•C4 ~t~~l(B(x 0 , Kr)) :::; (2Kr)d-•c4(Kr)d_, = C5 rd''>+d- 2•, where 0 5 > 0 is a constant. Consider a decreasing sequence of positive numbers Pk -+ 0 (k -+ oo) such that IL(B(xo, p~o;}) ~ Pk!l,.(p)-e for all k. We can also assume that Pl < min{r1,r2,r3 }. It follows now from (26.4) that
If k -+ oo this yields Since e is arbitrary this implies the desired result. We now proceed with measures which do not have local (semi-) product structure. We will first establish a crucial property of hyperbolic measures: they have nearly local product structure. This enables us to apply a slight modification of the above approach to obtain the desired result In order to highlight the main idea and avoid some complicated technical constructions in the theory of dynamical systems with non-zero Lyapunov exponents we assume that the map f possesses a locally maximal hyperbolic set A which supports the measure 1-'· Although the set A has direct product structure hyperbolic measures supported on it, in general, do not have local (semi-) product structure. For the general case see [BPS1]. Consider a Markov partition 'R. = {R 1 , ... ,Rp}. In order to simplify notations we set R~(x) = R,k ;1 (x) (the element of the partition vl=kfi'R. that contains x). We point out the followmg properties of the Markov partition. Given 0 < e < 1, there exists a set r c M of measure Jt(r) > 1-e/2, a.n integer
Relations between Dimension, Entropy, and Lyapunov Exponents
285
n 0 ;:: 1, and a number C > 1 such that for every x E r and any integer n ;:: no the following properties hold: (a) for all integers k, l ;:: 1 we have (26.9) JL~•>(R2(x))
c-Ie-k(h-e}
S
S ce-k(h+el,
(26.10)
c-le-l(h-e}
s JL~u)(~(x)} s ce-l(h+•)'
(26.11}
where h = h,_.(j) (the measure-theoretic entropy off with respect top,); (b) e-(d(•>+e}n
S p,~·>cs(x, e-")) S e-(d(•> -•)n,
(26.12) (26.13)
(c) define a to be the integer part of 2{1 +e) max{1/ ..\ 1 , -1/ Ap, 1}; then R:~(x)
R~,.(x) nA(x)
c B(x,e-") c R(x),
(26.14)
c s(x,e-") c R(x) nA(x),
(26.15)
Rg"(x) n A(x),
(26.16)
where the sets A(x, 2e-") and R~.. (y) n s(x, 2e-") are not empty; then (26.17) and Jl!~(y) C Q,.(x) for each y E Q,.(x); (e) there exists a positive constant D = D(f) < 1 such that for every k ;:: 1 and X E r we have JL~(R~(x) n r);:: D,
p,~(R2(x) n r);:: D;
(f) for every x E rand n;:: no we have
Property {26.9) shows that the Shannon-McMillan-Breiman theorem holds with respect to the Markov partition 'R. while properties (26 10} and (26.11) show that "leaf-wise" versions of this theorem hold with respect to the partitions ng and Rh. The inequalities (26.12) and (26.13} are easy consequences of the existence of the stable and unstable pointwise dimensions d{•) and d(u) (see Proposition 26.2). Since the Lyapunov exponents at J.l·almost every point are constant the properties (26.14), (26.15), and (26.16) follow from the choice of the number a indicated above. The inclusions (26.17) are based upon the continuous dependence of stable
Chapter 8
286
and unstable manifolds in the CH"' topology on the base point on A. Property (f) follows from the Markov property. For an arbitrary Cl+"'-diffeomorphism preserving an ergodic hyperbolic measure Ledrappier and Young [LY] constructed a countable measurable partition of M of finite entropy which has properties (26.9)-(26.17). In [BPSl], the authors also showed that this partition satisfies Property (e) and simulates (in a sense) Property (f). They used this partition to obtain a proof in the general case. We follow their approach. It immediately follows from the Borel Density Lemma (see Appendix V) that one can choose an integer n1 ~ no and a set f' c r of measure J.!(f') > 1 - E such that for every n ~ n 1 and x E f', (26.18) 1
J.!~·l(B(x, e-")
n r)
~
2 1
J.!i"'(B(u) (x, e-")).
2
(26.19) (26.20)
r
Fix X E and an integer n ~ nt. We consider the following two classes '!'(n) and ~( n) of elements of the partition 'R~~ (we call these elements "rectangles"): 'I(n) = {~~(y) c R(x) · R:~(y) n ~(n)
= {R:~(y) c
r # 0};
R(x): ~,.(y) n f # 0 and Rg"(y) n f' # 0}.
The rectangles in '!'(n) carry all the measure of the set R(x) n r Obviously, the rectangles in '!'(n) that intersect f' belong to ~(n). If these were the only ones in ~(n), the measure p.JR(x)nr would have the local direct product structure at the ~eve!" n and its pointwise dimension could be estimated as above. In the general case, the rectangles in the class ~(n) are obtained from the rectangles in '!'(n) (that intersect f') by "filling in" the gaps in the product structure. See Figure 21 where we show the rectangle R(x) which is partitioned by "small" rectangles R~~(y). The black rectangles comprise the collection 'I(n). By adding the gray rectangles one gets the direct product structure on the level n. We wish to compare the number of rectangles in 'I(n) and \V(n) intersecting a given set. This will allow us to evaluate the deviation of the measure JL from the direct product structure at each level n. Our main observation is that for "typical" points y E f' the number of rectangles from the class 'I(n) intersecting A< 8 l(y) (respectively A(u)(y)) is "asymptotically" the same up to a factor that grows at most subexponentially with n. However, in general, the distribution of these rectangles along A(•l(y) (respectively A(ul(y)) may "shift" when one moves from point to point. This causes a deviation from the direct product structure. We will use a simple combinatorial argument to show that this deviation grows at most subexponentially with n.
288
Chapter 8
Lemma 2. For each y E R(x) n f and integer n 2: n 1 we have
N(n, Qn(Y)) 2: p.(B(y, e-")) · 2Ce-Zan(h+e). Proof of the lemma. It follows from (26.17) and (26.18) that 1
2p.(B(y, e-n)) ::;'; p.(B(y, e-n) n r) ::;'; p.(Q.. (y) n r)
:::; N(n, Q,.(y)) max{p.(R): R E '!'(n) and R n Q.. (y) 1= 0}.
•
The desired inequality follows from (26.9).
Lemma 3. For p.-almost every y that for each n 2: n 2 (y) we have
E
R( x) n f' there is an integer n 2 (y) 2: n 1 such
N(n + 2, Qn+2(y)) ::; j{(•l(n, y, Q.. (y)). j{(u)(n, y, Q.. (y)). 2C2e4a(h+e)e4an"
Proof of the lemma. Since f' c r, by (26.17) and the Borel Density Lemma (applied to the set A= f', see Appendix V), for p.-almost every y E f there is an integer n 2 (y) 2: n 1 such that for all n 2: n 2 (y),
2p(Q,.(y) n r) 2: 2p.(B(y, e-n) n r) 2: p.(B(y, e-n))
2: t-t(B(y,4e-n-z)) 2: JL(Qn+2(Y)).
(26.21)
For any m 2: nz(y), by (26.9) and Property (d), we have
p.(Qm(Y)) =
L
IL(~:;!(z))
2: N(m, Qm(Y)). c-Ie-2am(h+e)_
R~:::(~)CQm(Y)
Similarly, for every n 2: n 2 (y) we obtain
where N .. is the nlliilber of rectangles R~~(z) E '!'(n) that have non-empty intersection with f. Set m = n + 2. The last two inequalities together with (26.21) imply that N(n+ 2,Qn+2(y)) ::;'; Nn · 2C 2e4 a(h+e)+4ane. {26.22} On the other hand, since y E f' the intersections R8n(y) n A(ul(y) n f and n A(n, y, Q.. (y)) 2: N ... The desired inequality follows from (26.22).
•
Relations between Dimension, Entropy, and Lyapunov Exponents
Lemma 4. For each x
E
t
289
and integer n 2:; n 1 we have
Proof of the lemma. Since the partition n is countable one can find points y; such Lhat the union of the rectangles Rgn(yi) is the set R(x) and thebe rectangles are mutually disjoint. Without loss of generality we can assume that y; E t whenever Rgn(y;) n t # 0. We have
(26.23)
By Properties (e), (f), and (26.10) we obtain that N(n
J.L~~)(Rgn(y,) nf) (•) () max{J.Lz (Rg~(z)) : z E A • (y;) n R(x) n r D
#
0}
- max{J.L~·>(R~n(z)): z E A(y;) n R(x) n r
#
0}
. uan(y·)) > ,y.,uo ' -
(26.24)
;::: nc-lea.n(h-c).
Similarly, (26.9) implies that N(n R(x))
'
cB(y,r)) = d. logr
r-+0
Relations between Drmension, Entropy, and Lyapunov Exponents
291
Clearly, JJ.(F') = JJ.(F) > 0. Then one can find y E F such that ~-t~•l(F) = p.~•l(F') = p.~•l(F'
n R(y) nA 0.
It follows from Frostman's lemma that dim 8 (F' n A(•l(y)) = d(•l.
(26.28)
Consider the collection of balls
8 = { B(z, e-m;(zl) : z E F' n A{•l(y), j = 1, 2, .. }. By the Bcsicovitch Covering Lemma (see Appendix V) one can find a countable subcover C C B of F'nA 0 one can choose a sequence of points {z; E F' n A L for each i, such that the collection of balls C = {B(z;,e-t'). i = 1, 2, ... } comprises a cover ofF' n A(•l(y) whose multiplicity does not exceed p. We write Q(i) = Qt, (z;). The Hausdorff sum corresponding to this cover is 00
l)diamB)a_, BEC
= :~:::e-t;(a-•J. i=I
By (26.27) we obtain 00
00
~::::e-t;(-e) i=l
: :; L
.zVC•l(t;, Z;, Q(i)). 4ce-at;h-4at;e
i=l 00
:::; 4CLe-aqh-4aqe L q=l
.zVC•l(q,z,,Q(i)).
i t;=q
Since the multiplicity of the subcover C is at most peach set Q(i) appears in the sum 2:::; t; =q .N< 8 l (q, z;, Q (i)) at most p times. Hence, L i
.N d(s) +d(u) r-+0 logr ..~ -n for ~t-almost every y E M. This completes the proof. • Let A. be a hyperbolic set for a C1+ 0 -diffeomorphism f on a compact smooth "lllanifold M. Assume that /lA. is topologically transitive. The set A. is called a hyperbolic attractor if there exists an open set U such that A. C U and f(U) c U. Clearly, A. = nn>o r(U). One can show that if A. is a hyperbolic attractor then wl~u; (x) c A. fur every X E A. (see, for example, [KH]). Consider the unique equilibrium measure p = fl."' on A corresponding to the function cp = J(u) (x). It is known that Jl."' is the SRB-measure (see [Bo2]). Obviously, d(u) =dim wi - hp.(/) -
= - 2:~=1
>.~q)
k~) >.~)
>.~q)
This implies that
. A> d lmH _ dimH J1. > _ m-
"~ L...J=l
k(j) >. (j) ~' 1-1 (q)
>.p.
Appendix V
Some Useful Information
1. Outer Measures (Fe]. Let (X,p) be a complete metric space and m a a-sub-additive outer measure on X, i.e., a set function which satisfies the following properties: (1) m(0) = 0; (2) m(Zl):::; m(Z2) if Z1 c Z2 c X; (3) m(.U Z;) :::; Em(Z;), where Z; C X, i = 0, 1, 2, .. •;:::o
;;:::o
A set E c X is called measurable (with respect tom or simply m-measurable) if for any A c X, m(A) = m(A n E) + m(A \E) The collection 2l of all m-measurable sets can be shown to be a a-field and the restriction of m to 2l to be a a-additive measure (which we will denote by the same symbol m). An outer measure m is called (I) Borel if all Borel sets are m-measurable; (2) metric if m(EUF) = m(E) +m(F) for any positively separated sets E and F (i.e., p(E, F)= inf{p(x, y): x E E, y E F} > 0); (3) regular if for any A C X there exists an m-measurable set E containing A for which m(A) = m(E). One can prove that any metric outer measure is Borel.
2. Borel Density Lemma (Gu]. We state the result known in the general measure theory as the Borel Density Lemma We present it in the form which best suits to our purposes.
Borel Density Lemma. Let A C X be a measurable set of positive measure. Then for p.-almost every x E A, lim p.(B(x, r) n A) = 1. p.(B(x, r))
r->0
Furthermore, tf J.t(A) > 0 then for each o> 0 there is a set t. c A with ~J-(t..) p.(A)- and a number ro > 0 such that for all x E t. and 0 < r < ro, 1 p.(B(x, r) n A) 2: /.L(B(x, r)).
o
2
293
>
Appendix V
294
3. Covering Results [F4], [Fe]. In the general measure theory there is a number of "covering statements" which describe how to obtain an "optimal" cover from a given one. We describe two of them which we use in the book. Consider the Euclidean space lRm endowed with a metric p which is equivalent to the standard metric.
Vitali Covering Lemma. Let A be a collection of balls contained in some bounded region oflRm. Then there is a (finite or countable) disjoint subcollection B = {B;} C A such that
UBcU.B;, BEA
i
where B; is the closed ball concentric with B; and of four times the radius. be a set and r: Z --+ JR+ a Besicovitch Covering Lemma. Let Z c r bounde.d function. Then the cover g = {B(x, r(x)) : x E Z} contains a finite or countable .mbcover of finite multiplicity which depends only on m. As an immediate consequence of the Besicovitch Covering Lemma we obtain that for any set Z c r and e > 0 there exists a cover of Z by balls of radius e of finite multiplicity which depends only on m.
4. Cohomologous Functions [Rl]. Two functions cp 1 and cp 2 on a compact metric space X are called cohomologous if there exist a Holder continuous function "1: X --+ lR and a constant C such that cpl - cp2 = "' - 1J 0 f + c. In this case we write cp 1 "' cp 2 • If the above equality holds with G = 0 the functions are called strictly cohomologous. We recall some well-known properties of cohomologous functions: (1) if cp 1 "'cp2 then for every x EX, I n-1 lim - L[cpt(x)- cp2(x)] = C;
n---too
n
k=O
(2) cp1 ,...., cp2 if and only if equilibrium measures of cp1 and cp2 on X coincide; (3) if the functions cp1 and cp 2 are strictly cohomologous then Px(cp1 ) = Px(cp2)· 5. Legendre Transform (Ar]. We remind the reader of the notion of a Legendre transform pair of functions. Let h be a strictly convex C 2-function on an interval I, i e., h"(x) > 0 for all x E 1. The Legendre transform of h is the differentiable function g of a new variable p defined by (A5.1) g(p) = max(px- h(x)). :tEl
One can show that: 1) g is strictly convex; 2) the Legendre transform is involutive; 3) strictly convex functions h and g form a Legendre transform pair if and only if g(a) = h(q) + qa, where a(q) = -h'(q) and q = g'(a).
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[Y1] [Y2] [Z]
(W]
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Index
Axiom A diffeomorphlsm, 228 u ( 8)-conformal, 230
coding map, 117, 229, 239 conformal map, 199 repeller, 199 toral endomorphism, 202 j-coordinate, 100 correlation sum, 174, 177, 181 cylinder (cylinder set), 100, 101
baker's transformation classical, 244 fat, 245 generalized, 244 skinny, 245 slanting, 246 basic set, 117, 190 Besicovitch cover, 31 Bowen's equation, 100 box dimension of a measure, 41, 61 of a set, 36, 61 BS-box dimension of a measure, 112 of a set, 111 q-box dimension of a measure, 58 of a set, 52, 62 (q, 1)-box dimension of a measure, 57, 63 of a set, 49, 62
0!-density, 45 dimension box of a measure, 61 of a set, 61 Caratheodory of a measure, 21, 67 of a set, 14, 66 pointwise, 24, 67 correlation at a point, 174, 178, 179 of a measure, 178 specified by the data, 181 Hausdorff of a measure, 41, 61 of a set, 36, 61 information, 186, 224, 256 pointwise, 42, 61, 143, 223, 254 BS-dimension of a measure, 112 of a set, 111 q-dimension of a measure, 58 of a set, 52, 62 (q, 1 )-dimension of a measure, 57, 63 of a set, 49, 62 pointwise, 58
CaratModory capacity of a measure, 22, 67 of a set, 16, 67 dimension of a measure, 21, 67 of a set, 14, 66 dimension spectrum, 32 dimension structure, 12, 66 measure, 13 outer measure, 13 pointwise dimension, 24, 67 chain topological Markov, 101 301
302 entropy capacity topological, 75 with respect to a cover, 77 measure-theoretic, 77 of a partition, 186 topological, 75 with respect to a cover, 77 equilibrium measure, 97 estimating vector, 134 expanding map, 146, 189, 197 expansive homeomorphism, 97 function stable, unstable, 105 geometric construction associated with Schottky group, 151 Markov, 118 Moran, 120 Moran-like asymptotic, 275 with non-stationary ratio coefficients, 121 with stationary (constant) ratio coefficients, 120, 152 regular, 122, 134 self-similar, 122, 133 simple, 118 sofic, 118 symbolic, 118 with contraction maps, 122, 150 with ellipsis, 138 with exponentially large gaps, 156 with quasi-conformal induced map, 145 with rectangles, 153 geometry of a