Luigi Amerio x B. Segre (Eds.)
Sistemi dinamici e teoremi ergodici Lectures given at the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Varenna (Como), Italy, June 2-11, 1960
C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy
[email protected] ISBN 978-3-642-10943-0 e-ISBN: 978-3-642-10945-4 DOI:10.1007/978-3-642-10945-4 Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg 2011 st Reprint of the 1 ed. C.I.M.E., Florence, 1960 With kind permission of C.I.M.E.
Printed on acid-free paper
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CENTRO INTERNATIONALE MATEMATICO ESTIVO (C.I.M.E)
Reprint of the 1st ed.- Varenna, Italy, June 2-11, 1960
SISTEMI DINAMICI E TEOREMI ERGODICI
P.R. Halmos:
Entropy in ergodic theory ....................................................
1
E. Hopf:
Some topics of ergodic theory ............................................. 47
J. Massera:
Les equations différentielles linéaires dans les espaces de Banach ........................................................ 113
L. Amerio:
Funzioni quasi-periodiche astratte e problemi di propagazione .................................................................... 133
L. Markus:
Sistemi dinamici con stabilità strutturale ............................. 149
G. Prodi:
Teoremi ergodici per le equazioni della idrodinamica ........................................................................ 159
A. N. Feldzamen:
The Alexandra Ionescu Tulcea proof of Mcmillan’s theorem ....................................................... 179
CENTRO INTERNATIONALE MATEMATICO ESTIVO (C.I.M.E)
PAUL R. HALMOS
ENTROPY IN ERGODIC THEORY
ROMA - Istituto Matematico dell' Università - 1960
1
- I -
P.R.Halmos
CONTENTS PREFACE CHAPTER I. CONDITIONAL EXPECTATION .Section 1. Definition Section 2. Examples Seotion 3; Algebraio properties Seotion 4. Dominated ccnvergence Section 5. Conditional probability Section 6. Jensen's inequality Seotion
7. Transformations
Seotion 8. Lattioe properties Seotion 9. Finite fields Seotion 10. The martingale theorem CHAPTER II. INFORMATION Seotion 11. Motivation Seotion 12. Definition Seotion 13. Transformations Seotion 14. Information zero. Seotion 15. Addit i vity Section 16. Finite additivity Seotion 17. Convergenoe Seotion 18. McMillan's theorem CHAPTER III. ENTROPY Seotion 19. Definition Seotion 20. Transformations Seotion 21. Entropy zero Section 22. Conoavity
3
- II -
P.R.H'llmos
Seotion 23. Addi ti vity Seotion 24. Finite additivity Seotion 25. Oonvergenoe CHAPTER IV. APPLIOATION Seotion 26. Relative entropy Seotion 27. Elementary.properties Seotion 28. Strong ,monotony Seotion 29. Algebraio properties Seotion 30. Entropy Seotion 31. Generated fields Seotion 32. Examples
4
- 1 -
P.R.Halmos PREFACE Shannon's theory of information appeared on th,mathematical soene in 1948; in 1958 Kolmogorov applied the new subject to solve some relatively old problems of ergodic theory. Neither the general theory nor its speoial applioation is as well known among mathematioians as they both deserve to be; the reason, .probably, is faulty oommunioation. Most extant expositions .of inflormation theory are designed to make the subjeot palatable to non,mathematioians, with the result that they are full of words like "souroe" and "alphabet". Suoh words are presumed to be an aid to intuition; for the serious student, however, who is anxious to get at the root of the matter, they are more .likely to be oonfusing than helpful. As for the recent munioa~on
~godic
applioation of the theory,the com-
trouble there is that so far the work of Kolmogorov and
his .sohool exists in Doklady abstracts only, in Russian .only. The purpose ·of these notes is to present a stop-gap exposition of some ·of the general theory and some of its applioations. While a few of the proofs may appear slightly
different from the oorrespon-
ding ones in the literature, no claim is
m~de
for the novelty of
the results. As a prerequisite) ' some familiarity with the ideas of the general theory of measure is assumed; Halmos's leasure
theory (1950) is an adequate referenoe. Chapter I begins. with. ,elatively well known faots about conditional expeptations; for the benefit of the reader who does not · know this technioal.prohabilistioooncept, several standard proofs are reproduced. Standard referenoe: Doob,
~tochastic
processes
(1953). A speoial oase of the martingale convergence theorem is proved by what is essentially L6vy's original method (Thdorie de
5
-
2 -
P.R.Halmos
~1addition des variabLes aLlatoires (1937)). The reader who kn ows the martingale theorem can skip the whole chapter, except possi b ly Section 9, and, in partioular"
equation (9.1).
Chapter II motivateeanddefines infor.mation. Standar.d reference: Khinohin, MathematicaL foundations of information theory
(1957). The ,mor.e reoentbook of Feinstein) Foundations of informa-
tion theory (1958}, ,is quite ,teohnioal, but highly recommended. The .chapter ends with Ii. pnoof :MoMillan f s theonem Gmean ,convergenoe}; the reader who knows that theorem oan skip the chapter after looking at .it .just long enough to absorb the notation. Almost everywhereoonvergence probably holds. ' A recent paper :by :Breiman (Ann. Math. Stat. 28 (1957) 809-811) asserts ,it, but that paper has an error j at the time ,these lines ,were written the correction has not appeara,p. yet. In any ·oase, for the ergodic application not even mean oonvergence is neoessary; all th,t is needed . isthe convergence of the integrals,whioh ,is easy to prove direotly. Chapter iIII studies entnopy (average amount of .infollmation)
j
.all .the faots .hene ane dir.eot :oonsequenoes .of the definitions, via the 'maohinery ,built up .in .the first two chapters. Chapter ,IV ,contains ,the application to ergodic .theory . In general terms, the idea is ,that information theory suggests a new invariant (entropy)ofmeasure~presenving transfonmations. The new invar.iant ,is shanp enough :to distinguish ,between some hitherto indistinguishable transformations (e.g., the 2-shift and the 3.shift). The ,original idea ,of using this invariant is due to K01mogorov ' (Do~lady 119 fi958) 861-864 and 124 (1959) 754-755). An improved ,version of the definition is, given .by. Sinai
(Doklady
124 (1959) 768-771l, who also oomputesthe entropy of ergod i c
6
- 3 P.R.Halmos automorphisms of the torus. The new invariant is in some resp ec ts not so sharp as older ones. Thus for instanoe Rokhlin (Doklady 124 (1959)
9BQ~983)
asserts that all translations (in oompaot a-
belian groups) have the same entropy (namely zero); he also begins the study of the oonneotion between entropy and speotrum. Muoh remains to be done along all these lines.
7
- 4 -
P.R.Hal mos
CHAPTER I. CONDITIONAL EXPECTATION SECTION 1. DEFINITION .We :shall work, throughout what follows, with a fixed probability space
(X, ~, . pl.
~
Here X is a non-empty set, a prpbabili ty measure on
is a field of subsets of X, and P is
-8 ., The
an abbreviation for "oolleotion
word "field" in these notes is of sets olosed under the forma-
tion of oomplements and countabLe unions". A probability measure ,on a fieid .of subsets of X is a measure P suoh that p(X) = .1 Suppose that
S
is a subfield of ~
and f is an integrable
real funotion an X. If Q(C)
for eaoh C in
& ,
= JC
f dP
then Q is a signed measure on (;
, absolute l y
ooptinuous with respeotto P (all, rather, with respeot to the restriation of
P to
~ ).
The Radon-Nikodymtheorem implies the exis-
.t e nce of an integrable funotion flf , measurable 13
S '
=J
fit dP C The funotion fit is uniquely determined (t o Q(C)
for eaoh C in
, such that
within a set of measure zero); its dependenoe an f and ~
i s in-
dioated by writing
The funotion E{r/!!) is oalled "the oonditional expectation of f with respeot to ~
". It is worth while to repeat the aharac-
teristic properties of oonditional expeotatibn; they are that (1. 1)
E(f/~)
is measurable ~
9
- 5 P.R.Halmos and (1. 2)
I C E(f/~)
=I
dP
C
f dP
for each C in ~ . SECTION 2. EXAMPLES. I f that is
(;=,(;
t;
is the hrgest subfield of
6 '
,then r i tseU satisfies (1.1) and (1. 2), so tltat E(f/~ )
= f .
This result has a trivial generalization: since f always satisfies (1. 2)
(fc
f dP =
Ic
f dP), it follcws that if the field
that f is measurable
13 ,
(2. 1)
E (f/ ~) = f
I;
is such
then
To look at the other extreme, let 2 be the smallest sub field of
~ , thH is the field whose only non-empty member is X . Since the only functions .measurable 2 are constants, and since the only constant (in the role of E(f/S )) that satisfies (1.2) is
Ic
f dP,
it follows that (2. 2)
E(r/2)
The constant
I
C
=I
r dP .
f dP is sometimes called the absolute (as oppose d
toconditionall expectation of
r , and, in that case, it is denc-
ted by E(f). Here is an illuminating example. Suppose that X is the unit square, with the collection of Borel sets in the role of
4
and
Lebesgue measure in the role of P . We Bay that a set in ~ is "vertical" in case its intersection with each vertical line L in the plane is either empty or else equal to X n L . The collection
~ of all vertical sets in ~
is a subfield of ~
10
. A function f
~
6
~
P.R.H!!.lmos i:sme!isur'lble
1$
i f and only if it does not depend on its second
(vertioal) argument; it follows easily that if f i~ integr!!.ble, .then E(f/~
SECTION 3.
AL~EBRAIC
lex, y) =
J
f(x, u) du .
·PROPERTIES. 'Conditional expectation is a
generalized integral and in one form or another it has all the
pro~
perties of an integral. Thus, for instance, (3.1)
E(l/~)
= 1,
W!,0l'e this equation, as well as '111 other asserted equations and inequalities involving oonditional expeotations, holds almost everywhere. (To prove (3.1), apply
(2~1).)
If f and g are integrable
funotions and if a and .b are oonstants, then (3.2)
"-ECaf+bg/~) =aECf/~) +bECg/(:;)
' (Proof: if C is in ~, then the integrals over C of the two sides
»
of (3.2) are equal to eaoh other). I f f:;: 0 , then ~(f/~) ~
(Proof,: i f C=
J
C
O.
E(f/!)(x) < Ot,then C is in
.h
~
and
f dP = 0; this implies that P(C) = 0 ). It is a consequence of
(3.3) that
(Proof: both If I
-
f ~ 0
and
1r:1
+ f ; 0, and therefore, by
(3.2) and (3.3), both E(-f/~) . ~ Eclfl/~) and E(f/~) ~ E(lfl /~) ) . Conditional expeotations also have the following multipli c!!.ti ve property: if f is integrable and if g is bounded !!ond measurabl e
11
- 7P.R.Ha.1mes
~, then (3. ·5) Sinoe the right side. of (3.5) is measurable ~ , the thing to prove is that (3.6)
J
c
E(f/b)gdP=J
C
fgdP
for eaoh C in ~. In oase g is the oharacteristio funotion of a
I; ,
set in
(3.6) follows from the defining equation 0.2) for
oonditiona1 expeotations. This implies that (3.6) holds whenever g is a finite linear oombination of suoh oharaoteristio funotiens J and henoe, by approximation, that (3.6) holds whenever gis a .bounded funotionmeasurab1e ~. SECTION 4. DOMINATED CONVERGENCE. The .usua1 .limit theorems for integrals also have their analogues for oonditional expeataHons. Thus i f f, g, and fn are integrable funotions, if Ifni ~ g and f
~
n
f almost everywhere,. then E(f It;)
(4.1)
-+ E(fl
n
G)
almost everywhere and, also, in the .mean. For, the. proof) . write.
g
n
= sup. (I f n - fl, If
1I.1.1
-fl,
If
n.+2
-fl,.·.. );
observe that the sequence{g } tends monotone1y to 0 almost every/l
where and, that g n ~2g, .. It follows that the sequenoe{E:(g . ' . n It
)}
is monotone decreasing and, therefore, has a limit h almost everywhere. Sinoe (4. 2)
Jh dP ~ J E (g/ ~,) dP = J gn dP ,
12
- 8 -
P.R.Halmos and since
J gn
dP'" 0, this implies that E(g/~) ~ 0
almost every-
where. Since, finally, (4.3)
the proof of almost everywhere convergenoe is complete. Mean oonvergence is implied by the inequality (4.4)
and the Lebesgue dominated convergence theorem. SECTION 5. CONDITIONAL PROBABILITY. If A is a measurable set (that is A is in ~ ) and i f f
= 0 (A)
(where c(A) is the charaoteristic function of A), E(f/t;)
= P(A/~)
~e
write
•
The function P(AI ~) is called lithe conditional probability of A with respeot to ~ ". The oharacteristic properties of conditional probability are that p('V ~)
is measurable ~
a,nd
J
o
for each 0 in
G. If
(5. 1)
P(A/t',) dP ::: peA
n 0)
A is in ~, then
l' (AI ~) =
0
(A) ,
and, in any oase, p(A/2) = P(A). For this reason the constant peA) is sometimes called the absolu-
13
- 9 P.R.Halmos te (as opposed to oonditional) probability of A. The oonverse of the oonolusion (5.1) is true and , sometimes useful. The assertion is that i f P(A/G) is the oharacteristic function of some set, say B, then A is in (and therefore B
= A).
To prove this, note that
JC and therefore peA
n
t;
dP
o(B)
= PCB
C)
= p(An
C),
n C) for eaoh C in ~ .' Since p(AI ~)
is measurable ~, the set B itself belongs to ~ . It is therefore permissible to put C
=B
and to put C
=X
- Bi it follpws that B
c:
A
and AC B (almost), so that B = A (almost). Just as oonditional expectation has the properties of an integral,
oondit~onal
probability has the properties of a probability
measure. Thus if A is
measurable set, then
Ii
and if ,{An} is a disjoint sequence of measurable sets with union A,
then P.(AI
e)
= L
n
peA II:':,)
n
SECTION 6. JENSEN'S INEQUALITY. A useful analytic property of integration is known as Jensen's inequality, which we now proceed to state and prove in its generalized (conditional) form. A real-valued funotion F defined on an interval of the real line is oalled convex if F (ps + qt) < pFeil)
+ qF(t)
whenever sand t are in the domain of F and p and q are non-negative
14
- 10 P.R.H'l.lmos numbers with sum 1. It follows immediately, by induotion, that if t l ,· •• ,t n are in the domain of F and Pi'" "P n are non-negative numbers with sum 1, then
(6. 1) Suppose now that F is a oontinuous oonvex funotion whose domain is a finite subinterval of
[0,00), suppose that g is a mea-
surable funotion on X whose range is (almost) included in the domain of F, and suppose that ~ is an arbitrary subfield of
-0
Jensen a inequality asserts that under these conditions F(E(gl ~ )) ~ E(F(g) / ~)
(6. 2)
almost everywher.e. Sinoe
is the limit of an increasing sequence
~
of simple funotions, and sinoe F is continuous, it is suffioient, to prove (6.2) in oase
where the summation field of
0 . If
extends over
the atoills of some finite sub-
g has this form, then
and
Sinoe E(F(g)/~) :; 2
A
p(A/~)F(t ), the inequality (6.2) is in
this oase a special oase of In the extreme oase,
A (6.1).
i::; -b ,
the conditional form of Jensen's
inequality reduces to a triviality (F(g) ~ F(g)); in the other extreme oase,
~:; 2, it becomes the olassioal absolute Jensen's
inequality
15
- 11 P.R.H~1mos
F(I SECTION
7.
g dP) S
I F~g)
dP .
TRANSFORMATIONS. Later we shall need to know the
effeot of measure-preserving transformations on oonditional expectations and probabilities. Suppose therefore that T is a measurepreserving transformation on X; this means that if A is measurable, ,
-1
then T
A is measurable and peT
-1
A) = peA) .
(For present purposes T need not be invertible). If
~ is a sUbfieU of ~ , then T-~~ -1
is the colleotion (field) of all sets of the form T
C with C in
~; if f is a funotion on X, then fT is the oomposite of f and T.
The basic ohange-of-variables result is that if f is integrable, then
Ic
= I. T--1 C
f dP
fT dP
for eaoh measurable set C. If, in partioular, C is in ~, then
I
-1
T C
E(fT/T
=
I c- f
=
I
-1
-1
dP
T C
~) dP =
=I
C
I
-1
T C
fT dP
\ E(f/~) dP
E(fl t;)T
1
dP . -1
Since both E(fT/T- ~) and E(fl ~)T are measurable T lows that
(7. 1) Since o(A)T
E(ft/T
= 0 (T
-1
-1
~)= E(fl ~)T .
A), this implies that
(7. 2) 16
~, it fol-
- 12 P.R.H'l lrr,os SECTION 8. LATTICE PROPERTIES. The
item of in te rest i s th e
ne~t
dependenoe of E(f/~) on ~ . The colleotion of all subfields of-6 has a reasonable amount of struoture;
it is partially ordered ( by
inolusion), and, in faot, i t is a oomplete lattice.
(The infimum of
two subfields ~ and ~ is just their interseotion, and their supremum
13 v ~
is the field they generate;
similar assertions hold
for the infimum and supremum of any family of subfields). It migh t therefore be hoped that the dependenoe of E(f! ~) on
t,
ex h i b i t s so -
me algebraioally pleasant behavior, such as monotony or addi t i v ity , Nothing like this is true. If,
for instance,
13
~ are subfields of ~ with 33.:; ~ ,
and
then the best that oan be said is that E(E(f/~)!03)
(8. 1)
= E(f/1l!)
and JIE(f / J3)1 dP ~ JIE(f/e,)1 dP.
(8.2)
To prove (8.1) observe that if B is in ~, then B is in
G,
and therefore
=JB
To prove (8.2), write B = B is in ~,
h:
ECf/i;)dP
E(f/~)(x) > O} . Since, c learly,
it follows that
J
B
E(f/13) dP
=
J
B
E(f/~) dP
k . It follows that i f
Observe that F n is in n ~ k , then P(F
n
- A) = f
and henoe that if F =
Fn
U n~k
P(F -
[1 - peAl F , then n
A) ~ P(F)€ 21
c; n ) 1
dP > P(F ) € n
J
- 17 P.R.Halmos This implies that P(B - A) ~ P(F)
< cS!2 , it follows that I f x is in B - F,
€
Sinoe P(B - p.) ~ P(B + A)
1 -
throughout B, exoept possibly in a set of measure lesa than S oall that
€o/2
+ Z/2 < 0 • Sinoe
€
j
€
re-
is arbitrarily small, it fol-
lows that P(A/ ~ ) oonverges to 1 throughout A, exoept possibly n
in a set of me~sure lees than 0
sinoe 8 is arbitrarily small, it
follows that p(A/~ n ) oonverges to 1 almost everywhere in A . This .. result applied to X - A in plaoe of A shows that p(A/l3 n ) oonverges to o(A) almost everywhere.
22
- 18 P.R.H~lmos
CHAPTER II. INFORMATION
SECTION 11. MOTIVATION. What is a reasonable numerioal measure of the amount of information oonveyed by a statement? How muoh information, for instanoe, do we get about a point x of X when we are told that x belongs to a subset A of X? It seems reasonable to require that the answer should depend on the size of A (that is, on peA)) and on nothing else. In other words, the answer should be expressed in terms of a function F on the unit interval; the amount of information oonveyed by A shall be F(P(A)). Two further reasonable requirements are that the funotion F be non-negative and oontinuous. Two experiments (or, alternatively, two statements) do not neoessarily yield more information than one. If, however, two experiments are independent of each other, then it is reasonable to expeot that the amount of information obtainable from the two together is the sum of the two separate amounts of information. If we perform two independent experiments, and if the result of the first tells us that x is in A and the result of the seoond tells us that x is in B , then we know that x belongs to a set (namely An B) of ~e~ sure P(A]P(B) (by independenoe). The reqUirement of additivity implies, therefore, that the funotion F should satisfy the funotional equation F(st)
= F(s)
+ F(t)
throughout its domain, It is well known, and it is easy to prove, that the oonditiDDF imposed on F uniquely oharaoterize F in the interval (0,11
23
J
to wi-
- 19 -
P.R.Halmos thin a multiplioative oonstant. Indeed, two induotions (one for the numerator and one for the denominator) show that F(t r ) never r is a positive rational number. Continuity result whenever r is a F (e -,r) = r F ( e
-1
positl~e
= rF(t)
yi~lds
real number. put t
1 - ~ e
whe-
the same to get
) , or, in other words,
Conolusion: exoept for a constant factor, F(t)
=-
log t; the a-
mount of information oonveyed by the assertion that x is in A is satisfaotorily measured by - log peA).
SECTION 12. DEFINITION. As the preoeding discussion indicates, the mathematioal model of an experiment with a finite number of possible outoomes is a finite partition of X into measurable sets, or equivalently, and from the point of view of the intended applications more elegantly, a finite sub field of ~
The amount of in-
formation oonveyed by one of the possible outcomes of an. experiment is a number depending on that outoome; in other words, the· amount of information is a funotion of outoomes (associated with some experiments). These oonsiderations motivate the following definition. It
a
is any finite subfield of '"
assooiated with atom A of
a.
a is
, the information funotion I(a)
the funotion whose value at eaoh point of e!l.ch
is - log peA); equivalently
(12.1)
I(
a) =
~A
0
(A) log peA) .
Conditional information is a natural and useful generalization of this conoeptj it is obtained by using conditional probabilities instead of absolute probabilities. Explicitly, i f o'and t) are subfields of
.-6 ,
with
a
finite, then, by definition,
24
- 20 -
P.R.Halmos (12.2)
Observe that I( a!~) is always non-negative. The oonneotion between oonditional information and absolute information is that ( 12.3)
I (
see (5.2). The funotion
a12),
a.)
= I(
j
Qv~, a'nci, in
I(al!;) is measurable
partioular, I( a}is measurable
a.
SECTION 13. TRANSFORMATIONS. If T is a measure-preserving
tr~ns-
t'ormation on X , then
the proof is a straightforward applioation
(7.2). It follows in
~f
particular, that
SECTION 141. InFORMATION ZERO .. (14.1)
I(
rra. c t
Q.! ~) = 0
The proof 1ei &&Sedo'll ',65. U and
1(2)
=0
then
.
th& ~.C()nv:.ent,i.Qn
log t = 0 . It follows in particular, with (14.2)
,
0..=
tha.t i f t = 0, t he n 2 and
~ = 2, that
.
These equations are in harmony with the intuitive meaning of information. Thus, for example, .cU.~) Ilxpresses that i f before some partioular
experim~nt
experiment oan veyed by the
is perfor,ed r mere
posslb~r
outoo~e
is '
reveal,
. *~~~ ! ~h,
oertaJnly ~ z~ro.
25
IS
already known than the
a.ount.gf information oon-
- 21 P.R.Halmos The converse of the conclusion (14.1) is true; the assertion is that i f I(
(i/f;) = 0 almost everywhere, then
(14.3) Indeed, i f I( (l/~) = 0 almost everywhere, then c(A) log P(A/~) = 0 almost everywhere for eaoh atom A of
a . This
implies that P(A/
t,)
is a oharaoteristio funotion and henoe (by (5.3)) that A is in (;
.
SECTION 15. ADDITIVITY. Conditional probability (of a set with respeot to a field) is algebraioally well-behaved as a function of its first argument and analytioally well-behaved as a funotion of its second argument. These faots are refleoted by the behavior of oonditional information. If
!l,
~ and
G
a
are fields suoh that
and ~ are finite,
then I(
a.>'7;) /~)
= I( 'd/~)
The left side of (15.1) is symmetrio in
+ I(Q./ ~vt;
a,
and
) .
'tJJ , whereas the
r i ght side is apparently not. _This is no oause for alarm;
(15.1)
i s merely one of two equations, whioh between them describe the who-
~= 2, then (15.1) becomes
le (symmetrio) truth. If (15.2) if
I(
rtv'i'J)
= I(lJ) + I(Q../1j)
QC ~ , then (15.1) beoomes
If, in partioular, (15.4)
(;. = I (
Q
a , then V
13 /Ov)=
I (
26
J3! Q)
- 22 P.R.H'l.lmos
CtcTj ,
If, finally,
then
this follows from (15.1) and the faot that I(~/a..V~
) is non-
negative. The proof of (15.1) is the following oomputation: I(
Qv1:J1 ~) = - rArB o(A)o(B) log P(A n BI ~) A) 0 ( B) log ( P ( Bib
= -
~ AIB
0 (
=
~A
A) IB log P (B I ~)
0 (
-
)
P(AfiB/G)
IA 0( B)
ptA (f BI t,) P(BI ~)
SECTION 16. FINITE ADDITIVITY. The following assertion is a
13
useful oorollary of (15.3). If
I(
V._n
L-1
'l1
·(Ji)=~
The proof is induotive. For n I(
13 ) l
= I(
are finite
~o = 2,and n= 1, 2, 3, ... ), then
fields (with (16.1)
1'J 1 , •.. ,1O n
0,
131 12).
n
k=1
I('t>IV
= 1,
k
k-1
i=O
1).) 1
the assertion reduoes to
For the induotion step, use (15.3) as foI-
lows:
27
- 23 P.R.Halmos n+1
I(
n
Vi=l 131 ) = I( V1=1 'l3 i =
I(
V1:1
V 1)n+1)
lj1) + I( ijn+1/
V 1:1
1'0 1 )
An important speoial case of (16.1) is obtained as follows. Suppose that
a,
,6
is a f1nite subfield of
and that T is a mea-
sure-preserving transformation on X • For each positive integer n ,
13 i
"rite
= T-(n-i),., IN
( i = 1, ... ,n, ) and apply (6 1 . 1). The re-
sult is
I
n ( Vi=1
T-(n-i) .... ) = I(T-(n-1)".) vc. I.N n
+ ~k =2
>1
for n
I (T
-(n-k)
a
k-1
I
V i= 1
T
-(n-i)h IoU
)
. This can be rewritten in the form I(V n - 1 T-i(t) = I(T- Cn -l)Q..) i=O +
I(
n-1
V 1 =0
T
Ln -1 I ( T- ( n - k - 1)
-1
k=l
av)
aI V
k T - ( n - 1) i=1
a ).
a )Tn-1
=I(
+ ~n -1 I ( (L I k=l
V
k T- ( k - 1+ 1) i=l
a.. } Tn - k -1
or, equivalently, (16. 2)
n-1 I(
V 1=0
= I( ~)T
+
n-1 };k=l
n-1
I ( Q, /
V k T-iet... )Tn - k - 1 i=1
whenever n > 1. [The preoeding oomputation made use of the faot that
28
- 24 -
P.R.Halmos
for any oolleotion
E.,
To prove this, observe that sinoe T
-1
(V 6 )
V here
of sets;
'
GC Vt
denotes the generated field. so that T- 1 G C
-1
f...
C
V (T -1 C).
Sinoe the oolleotion of those sets A for whioh
V(T- 1 G )
V G );
is a field, it follows that
For the reverse inolusion observe that T
to
T- 1 (
T
-1
A
belongs
is a field, it follows that
and the proof is oomplete. If, in partioular, ~ and
t;
!lore fields, then put
G = ~U t.
and apply (16.3); the result is that
) =
-1
T
a., v
[The similar faots about infinite suprema are proved tha same way] .
SECTION 17. CONVERGENCE. If { ~n} of sub fields of {) , and if (17.1)
I(a/~n)
~
~= I (
is an inoreasing sequenc e
then Vn I:; n ,
(1;
t, )
almost everywhere. The proof of this assertion is immediate from the definition (12.2) and the oonvergenoe theorem (10.2). For some purposes oonvergenoe in the mean is more useful than almost everywhere oonvergenoe. Convergenoe in the mean is olosely oonneoted with uniform integrability. Reoall that a se-
29
- 25 P.R.H'l.lmos quen0e {fn} of measurable funotions on X is uniformly integrable if
tends to 0 , as t
~
Ol
,
uniformly in n
The faots are these. If
f is an integrable funotion and if {f } is a sequenoe of integran
ble funotions suoh that fn ~ f
in the mean, then {fn} is unifor-
mly integrable. If, oonversely, f is a measurable funotion, is a uniformly integrable sequenoe, and if f re, then f is integrable and f
n
~
n
~
{fn}
f almost everywhe-
f in the mean. In view of these
faots, the way to prove that (17.1) may also be interpreted in the sense of mean oonvergenoe is to prove uniform integrability. Note that even the integrability of one I( a.,/~) needs proof; there is no obvious boundedness to invoke. UNTFORM INTEGRABILITY THEOREM. If N is a positive integer, then the oolleotion of all funotions of the form I( (J.,/~) (where
0....
is a finite subfield of ~ with not more than N atoms and ~
is an arbitrary subfield of ~ ) is uniformly integrable. PROOF. Take N, (t, and ~
!HI
desoribed, and let rand s be
positive numbers, r < = s • Write D If A is an atom of
= h:
r ~ I(Q.,/t;)(x) ~ s } .
(L, thsn I(a/~) =
log P(A/~) on A . It
follows that if CA then
AnD =
belongs to
t,
A
n
= h:
r ~ - log P(A/~)(x) ~ s } ,
CA' Sinoe P(A/
G)
is measurlJ.ble ~ , the set CA
for eaoh A . Sinoe on CA log P(A/~) < - r
30
- 26 P.;; .H a llllo s or P(A/~) < e
-1'
it follows that
= Ic < = e
IUl/C) < s on
Since
Sum over all the atoms A of
D
Put
=t
l'
(17.2)
+ n , s
J {x:
I
< e -1' p(C A) =
-1' < dP ~ sP(A fl D) = se
a.. to
get
I( a.!~) dP ~ Nse
=t
_1'
p(A/~) dP
D ,
J n 1«1/0) A D J
A
+ n - 1 (n
((1 I ~ ) (x) ~ t}
= 0,
l'
1, 2, ... ); it follows that
aI ~) 100
I(
0 funotion;
and
F(O)
=0
, then
F
is a oontinuous convex
it follows that
F(E(p(A/~)/15); E(F(P(A/~)lra)
(22.2)
for eaoh atom A of , a,. By (8.3) the left side of (22.2) is equal to F(P(A/'b
»;
this implies that
To get (22.1), ohange sign and integrate; of. The speoial oase
~
(19.5).
= 2 is worthy of note :
(22. 3)
35
- 31 P.R.Halmos
a,
SECTION 23. ADDITIVITY. If
~
, 'loud ~
are fields such
Q., and ~ are finite, then, by (15.1),
that
It follows that
(23.2) and, in partioul!l.r,
Further speoial oases: if
a c~,
then
H«(lv'tJ/t;) =H(~/~) , H(a,v~/0) = ii(1Jllj) ,
H( ct v 1f> 1 I f - B
Thirdly, in B there holds sup S f I': 0 and, in particular, f = n + S f < 0 or, in other words, f = o. O. Lemma 1 .is thus seen to furnish n a simple proof of (37). However, we .have mentioned lemma 2 because
71
24 -
~
E.Hopf it is a handier form of (37). The following lemma is a known consequence of (37) but it is more easily derived from lemma 2. It deals with the quotients
(38)
q (x) n
= qn (Xi
f, p)
S f(x) n S p(x) n
=
n > 0
Note that, for any constant UJ,
LEMMA 3. If f
e
L, PEL, P
~
0 then
sup jqn(Xi f, p)j O holds almost every.where in the set where p > O. PROOF. As jqn(Xi f, p)j < qn(Xj
jfj, p) holds it suffices to pro-
ve the lemma for the case that f
~
0, and without the absolute va-
lue sign. Let A
=
[xjp(x) > 0, sup qn(Xj f, p) n>O
= 00]
•
By virtue of (39) sup qn(Xi g - UJP, p) > 0 n>O holds everywhere in A, for any constant UJ > O. In other words supS fl (x) > 0 n>O n
f
I
-
f - UJp
must hold in A. Hence, by virtue cf lemma 2,
I A(f From f
-
- UJp) +LU A
-
UJp)
+
~ 0
UJp < f, f
~ 0
it follows that (f
UJ
IAp
~
Therefore
IAf
+
I_f A 72
=
If .
-
wp)
+
< f
- 25 E.Hopf As w > 0 is arbitrary and as p > 0 in A it follows that A has measure zero, and lemma 3 is proved. For the following lemma of Chaoon and Ornstein we give an abbreviated proof. LEMMA 4. If f , L, P € L , p lim n~
~
0, then
Tnf(x)
=
S p(x}
0
n
holds almost everywhere in the set where p > O. PROOF. It suffioies to prove this for the oase that f a number
€
~
O. Piok
> 0 arbitrarily and oonsider the functions g
o
= f
whioh satisfy the relation (40 )
= Sntl I, I = identity. We need merely show that, in n almost every point of the set [p » ol, gn < 0 holds for all suffi-
since TS
oiently large n or, in other words, that!e n converges where en . is the oharaoteristio function of the set [gn ~ lation e nt1 gntl
t
= gntl
we infer from (40) and (33) that
On adding up the resulting inequalities from n
fg
t
n
t
€
0]. Using the re-
nt e i ~ go
JP !1
J
and hence
73
=0
on ,
- 26 E.Hopf -1
€
Jgo+
.
Lemma 4 is herewith proved. The next lemma oontains the prinoipal part of Chaoon's and Ornstein's proof of the ergodio theorem. LEMMA 5. Let f
~
L, PEL, P
O. If in ee.oh point of a set A
~
there holds p
> 0 ,
and
W
qn (x; f, p) < a < b < lim qn (x; f, p)
where a, bare oonstants then A has measure zero. PROOF. Write qn (f) for the quotients qn and note that, in oonsequenoe of part of the hypothesis, sup q (f - bp) n>O n
lim q (f - bp) > 0 n
~
holds in the set A. Choose
E
> 0 arbitrarily and apply lemma 1
to f-bp in plaoe of f. Write the h of this lemma in the form h-bp.We infer from a) that (41)
< (f -
b )
P
and from S) that
JA (h
(42)
- b) P
0
within A. Consequently, lemma 1 may be applied to ap-h in plaoe of the f there. write the new auxiliary funotion h of this lemma in the form ap-f'. Then it follows from y) that (44)
or
J(ap-f') < J(ap-h}
J(f'-ap) ~ J(h-ap)
and hom (I.) that
(45)
or
(f'-ap)
+
~ (h-apr
+
and, finally, from 0) that (46)
We now have a set of inequalities involving beside the given f two auxiliary h, f'. h can be eliminated from them in the following way. Split the integrands in the seoond inequality (44) into positive and negative parts, (47)
< J(h-ap)
-
+ J{(f'-ap)
+
+
- (h-ap) }
By virtue of (45) the secQnd integrand on the right is
~
O.
(47), therefore, stays valid i f the inte.gral is taken over A.
75
- 28 -
E.Hopf
(h-ap)
+
~
h-ap = h-bp + (b alp > - (h~bp)
+ (b-a)p .
Hence the right hand side in (47) is < J(h-ap)
+ J (fl-ap) A,
+
+ J (h-bp)- - (b~a) J p A
On applying (43) to the first term,
A
(~6)
to the seoond and (42)
to the third term we obtain that (48) By the same reasoning ae above (theuee of lowe that qn(fl) and qn(h) have the same
fi)
of lemma 1) it fol-
upp~r
and the same lower
limit in almost each point of A. Thereby the following result is obtained. If f satisfies thE! hypothesis of lemma 5 then, to any given
E
> 0, there exists a function fl whioh satisfies the same
hypothesis and, in addition, the inequality (48). This result may in turn be applied to fl and we get a function fll whioh satisfies the hypothesis of the lemma and (48), with f, fl replaced by fl, fll, and so forth. On writing these inequalities underneath eaoh ther and on adding up the first n 'of them
we
find, on neglecting
the non-negative integral remaining on the lett, that n ,{(b-a) J
A
p ..
2d < J(f..,ap)
must hold for any integer n. Hence
Lemma 5 follows from this since E was arbitrary and since a < b, and p > 0 in A.
76
0-
- 29 E.Hopf COMP~E~ION
OF ·THE :PROOF ·OF ;THE ·SEOOND ERGODIC 'THEOREM. 'We pro.
ve first that the qnhave a finite limit almost .every"here in the set[p >0]. We may suppose that f ~ ·0. Lemma 3 ,implies that the finite or infinite limit if it
e~ists
must be finite almost everY-
whene. So "e need only pnoveexistenoe.Within .the ,part ·of the set
.h
> 0) in "hioh
i ~ T p
00
a
,plies that the senies
this is obvious beoause ,lemma 3 .then im-
0), ho"ever, the positivity of the ,measure of
the set in "hiGh .lim S
n-k
f'
sn-k p'
=
>0 .
= -
k
..
From this formula and from the previous result as in ·(49) it tollo"s no" that the q
n
ap~lied
to fl, p'
have a finite li.italmost eve-
ry"here .in the set (49). The seoond ergodio theorem is hereby .oom·plately proved.
77
- 30 E.Hopf 1d. CONSERVATIVE AND DISSIPATIVE PART OF A MARKOV PROCESS. Consider again a positive linear operator T from
L~
to
L~
with a. norm IJTII = 1. We now turn to the simpler question of convergence or divergence of the infinite series Scof =
co i L T f o
f
€
L~
DECOMPOSITION. X is the union of two disjoint sets, the convergence set Xc and the divergenoe set Xd , with the following properties. If P f v ~ rywhere
L~,
P ~ 0, P > 0 in Xd , then Sco P
in Xd in the sense of the measure
= co
holds almost e-
~
the series Scof oonverges absolutely almost everywhere in Xo' PROOF. Let q > 0, q ELand denote by Xd and X;, respeotively, the sets where Sco q
= co,
< co . Consider a funotion p with .the properties
stated .above and apply lemma 3 to the quotients Snq/SnP' As p> 0 and Sco q = co holds in Xd it follows from this lemma that ScoP = co must hold a.e. in Xd ' Now let f ( L~ and 90nsider the funotion p' = q + I f I € L~, p' > O. Apply the same lemma to the quotients S p' /S q. As Scoq < co holds in X it follows that also ·S~p I < co n n o w holds a.e. in I . The sam, holds therefore for scolfl. The rest of o
the lemma follows from Ifnfl
~ Tnlfl.
This result shows that within X the ergodio theorem is trio vial and uninteresting. Both denominator and numerator oonverge separately a. e. inX o ' It is possible to separate Xd from Xo in suoh a way that the operator T somehow splits .into two operators one of whioh aots on and produoes funotions that vanish .in Xo whereas the other does the same relative to Xd? In other words, is the implioation (50)
f = 0
a. e. in
Xo
78
'9
Tf = 0
a. e. in Xc
- 31 E.Hopf valid? And is the analogous implication valid relative to Xd? (50) means that the values of Tg, for general g, wi thin Xc do not depend on the values of g within X . From the representation ford
mula
J Tf A
d~ =
J
X
P(x,A)fd~
we easily infer another equivalent of (50): P(x, X) = 0 holds a.e. c
in X . d
(50) is valid [14] and its proof is given below. However, the other implication regarding Xd does not generally hold good. It, is certainly valid if T is generated by a
on~-to-one
point transforma-
tion but in the other extreme case where T is a completely continuous operator the resufts of Yoshida and Kakutani [20] show that it is invalid or, in other words, that Xc has an influence on Xd . PROOF OF (50). We may assume that (51)
f ~
0 .
Consider a fixed q > 0, q , L . By the deoomposition theorem, J.L
a.e. inX d
(52)
q and that Note that S q = S n ntl TSnq = S q - q < ntl
S~q ~
.
On letting h (x)
n
= inf
we can infer from (51),
(55)
(f(x), S q(x» n
(52) and the premise of (50) that
o,Shff n
79
- 32 E.Hopf holds in Xc as well as in Xd . We show that a.e. Th h
In fact
f
n.
< h
n+l
f
~
n
~..., "?'
t
Tf •
Thn < Th
(Tf - Thn)d~ = f T(f-hn)d~ < X X
n+l
f
~
Tf . and, by virtue of (33 ) ,
(f-hn)d~,ltO .
X
almost everywhere. This holds for any f whioh satisfies the hypothesis and, therefore, also for
€
-1
f,
€
> 0 ,
almost everywhere. This holds for any f which satisftes the hypothesis aM, therefore, also for £-I f , e: > 0,
As Sooq
oontain a wandering set of positive measure. It is olear that these five steps make up the oomplete proof of the theorem. We merely prove d) and e). Reoall that the assooiated opera-
f
tor on set funotions is T¢(A) = ¢(TA), A' 'Tf'(TA)
=
JA
Tp d~
if
?T(B) =
J
and that pd~
B
where T is given by (57). The left hand identity holds for all pon
wers T , n (58)
0, too. Henoe
~
00
i
L ?T(T A) = J o
A
*
,
To prove d) take for A a wandering set such that D is the union i of all its images T A, i < O. As these images are pairwise disjoint it follows that the left hand side of (58) is .::. ?T(D*) .::. ?T(X)
0, such that SooP is bounded in B.On applying (58) to B we infer that
(59) 82
00,
- 35 E.Hopf co
Now let On;:
U TiB
and note that B ;:
n
virtue of (58), co
.
7T(On) < 2: 7T(T 1 B) n
oan be made arbitrarily small as n gets large, oertainly < 7T(00) for some n ;: k.
Oonsequently~
00 - Ok has positive measure
same must be true for at least one set A' ;: 0.
~
. The
0i+1' i ;: 0, ... , k-l.
1
Now observe that m
T A' ;:
m
>
°,
and that these sets, m > 0, are disjoint with A'. As is wellkhown this (and the assumption that T is one-to-one) implies that A' is a wandering set. The proof of the theorem is thereby finished. For the validity of the theorem the hypothesis that the mapping is one-to-one is absolutely essential. Recall again that the sets Xc' Xd are determined by a Markov operator in the first place. Now, if the mapping is one-to-one there is only one Markov operator which can be reasonably connected with the mapping, and so it is not surprising that Xc' Xd can be characterized by the mapping alone. The situation is entirely different in the case of a many-to-one mapping, We have seen above that, in general, the representation of such a mapping by a Markov operator is no longer
uni~ue.
That this
ambiguity has an influence on the sets Xc' Xd is clearly illustrated by the following example. Let X be the set of all natural numbers, x ~
::z
1, 2, . . . . Let
be the a-finite measure which is one for each such x. Oonsider
the point mapping x >1 , x ;: 1 ,
83
- 36 E.Hopf ~
of X onto X. 1 consists of all subsets of X, T
-1
f
only of those
subsets which contain either none of the two numbers 1, 2 or both. A Markov operator associated with T is of the form (57) where r
~
0
is such that (26), the conservation property of Markov operators, is satisfied for each f 00
~ -00
L ,
~
f1-
00
f(x) ::
This requirement is .
2: r(x) f(Tx) .
-00
read~ly
r(1) + r(2) :: 1 ,
seen to be equivalent to the relations r(3) :: r(4) ::
=1
.
So we have infinitely many Markov operators (57). There holds (60 )
n
n
T f(x) :: r (x) f(T x) , n
r
:: r(x) r(Tx) ... r(T
n
n-l
x).
First case. r(1) :: 1, r(2) :: O. We distinguish between the two parts (61)
[x Ix >
1J
:: 1, Soof ::
00.
[xix:: 1),
of X. In the first set, r
II
f(l). However, in the se-
oond set, r (x) contains, for all large n, the factor r(2) :: 0 . n
Therefore Soof is finite. It follows that the first set is X
d
and
the second X . c
Second case. r( 1) < 1. No matter what x we start from the factors of rare:: r(l) from a certain place on. Hence, n
rn(x) < C(x)r(l)
n
and, oonsequently, Xo:: X, Xd:: O. Note, by the -1
way, that the unique r(x) which is T
f
-measurable, r(l) :: r(2),
comes under this seoond case. It is instructive in this connection to consider another example in whioh the spaoe X is not oountable. This example was briefly mentioned already
x is an angle variable mod. 27T and the
84
- 37 E.Hopf mapping is X'
= Tx = 2x mod 2n .
We found that each measurable function rex) 2 0 with satisfies the relation r(x) + r(x + n)
=
2
furnishes a Markov operator T , Tf(x) = r(x).f(Tx) , for the mapping T. In order to study the behaviour of the iterates Tn of the operator we begin with the simplest case r already mentioned that, for g € L
Ji.
= L 1,
lim
1 n-1 v L g(2 x)
n~CD
n
= 1.
It was
'
the limit
0
exists almost everywhere. We need the fact that the limit function g(x) is constant a.e., and that the constant equals g(y) dy . The simple reason for this is : For almost all x there holds g(2x)
= g(x),
g(2 k x)
= g(x)
where k .is an arbitrary positive inte-
ger. The Fourier expansion of g must therefore reduce to the constant term. Now we return to our question regarding the associated operator rf(x)
= r(x).f(2x)
where rex) is any admissible function> O. X is the set on the circle in which every series CD
i
L T p(x) o
85
- 38 E.Hopf 1
P > 0, PEL, oonverges a. e .. Now, n
= r n (x)p(2
T p(x) and ro(x)
= 1.
n
x) ,
= r(x) ... r(2
r n (x)
n-1
From the immediately preoeding result, g
it follows that log
x)
= log
r,
a.~.
1
n
V r (x) n
21T'
21T
f l o g r(y)dy . 0
As the log is oonvex the right hand oonstant is ~ log
f-
1
21T
f 21T 0
1
r(y) dy
and equal to it if and only if r is oonstant a.e .. The last named expression has the value zero as follows readily from the admissibility relation for r(x)-. Henoe we infer tha.t, for almost every x,
r'--r-
n
has a limit if r
=1
n-=-(x-:)
< 1 unless rex) is oonstant a.e. (whioh oan happen only
a.e.). If we oombine this with the faot that the set xo
is essentially independent of the ohoioe of the funotion p, p > 0, 1 n p , L , (we may take p = 1, T P = rn) we obtain the following ourious result. Any admiss-ible- funotion r, r > 0, rex) + rex + 1T)
= 2,
furni-
shes a Markov operator "assooiated" with the mapping. In the oac e where r
=1
a.e . we have X
o
=0
funotion "one" then we have X
o
but if r is ~
n~t
equivalent to the
Xl
We had notioed this faot already in 1946. There is, of oourse, the question wether there is an "opt"imal" Markoy operator whioh desoribes the future behaviour of the iterates of a many-to-one mapp~ng
better than the other assooiate operators. On oomparing the
86
- 39 E.Hopf results of the two examples mentioned here we can see that a generally valid answer cannot be trivial. However, the beautiful applications of information theory to ergodic theory which Kolmogorov has initiated and to which we were introduced in this Seminar by Paul Halmos may throw light on this question. Let us finally return to the general second ergodic theorem itself. We wish to conclude this section with the question (not answered as yet) about the nature of the limit function lim q (x). This n
question is, of course, interesting only in Xd . If we operate in Xd only, more precisely, if we confine ourselves to functions f vanishing in X then (50) has the following c
implication. Within Xd , T represents again an operator of the same kind! positive, linear and of
L~
- norm
~
1 within Xd' Furthermore,
if T is a Markov operator in X so is T in Xd' We may therefore assume, without loss of generality, that in
X if
P > 0 .
We need the notion of the dual operator T associated with the given operator T, (D)
explicitely,
'"Tg(x) = fp(x, dy..)g(y) . '"T is actually an operator from
~-equivalence
'"
class of bounded func~
tions to again such a class [14). T is positive, and T1 = 1 holds if T is a Markoff operator. We assume here that T is a Markoff operator (this property is used in the proof of one of the subsequently mentioned facts).
87
- 40 -
E.Hopf A bounded and
~-measurable
funotion hex) is said to be inva-
riant under the Markoff prooess if it is invariant under T, (62)
Th = h
(up to a set with
~
= 0).
The theory of these invariant funotions
was developed in [14], however, under restrotion to bounded funo,.. tions. The main results are the following (see [14], §9). The invariant funotions form an algebraio field. Also, Ihl is invariant if his.
If h is invariant T(ll w)
( 63)
there holds (almost everywhere)
t~en
=h
T(w)
for every bounded and measurable function w(x). Conversely, of oourse, the latter property of h implies invariance of h since T1
= 1.
The proof of (63) is given in [14], §9 . The property (63) is oompletely equivalent to the following property of the original operator T :
=h
(64)
T(h.w')
T(w')
holds for every
~_integrable
funotion w'(x). The equivalenoe of
the two properties follows easily by means of the duality relaHon (D).
Let us now return to the question of the limit funotion f* = lim
S
n
f
n-+oo
of the general ergodio theorem. It has not yet been proved that this limit funotion is invariant and that it is independently charaoterized by the oondition that pf* is ~~integrable and that
88
41 -
~.
E.Hopf holds for every bounded and invariant funotion h. We repeat that this conjecture is stated under the hypothesis that T is a Markov
0-
perator. In order to prove it is first of all necessary to extend the notion of invariant function from bounded functions to functions h' for which
JIh 'I
pd,u
is finite. This should not be difficult. We also would like to call attention to another way of proving the general ergodic theorem, namely, by generalizing a simple trick which F. Riesz onoe introduoed to prove the classical mean ergodic theorem. Consider first functions of the form Tg - g. The funotions h which are orthogonal to all of them must satisfy the relation
o = J(Tg
- g) hd,u
'"
= fg(Th
- h)d,u
for all g or, in other words, h must. be invariant, Th = h. Show that every f can be approximated by a sum of a function
~g
- g
and an invariant function h. Then prove the ergodic theorem in the two simple cases 1) f
= Tg
- g, 2) f = h. In the first case use
lemma 4 (with g written in place of f) to show that
S (Tg - g)/s p~o. n n
In the second case use t.he relatibns i T (h.p)
=h
i
T P
which follow by repeated application of (64). In this case there would hold
for all n. Finally, it must be shown that, for a function f' of
89
- 42 -
E.Hopf small norm, the funotion lim sup n~oo
is also small exoept in a set of small measure. This oan be inferred from our maximal ergodio lemma (lemma 2). This proof of the ergodio theorem, if oarried through oompletely, would not only prove the existenoe of the limit funotion but also its invariance.
90
- 43 E.Hopf 2a.
TWODIMENSIONAL~¥PERBOLIC
GEOMETRY.
The well-known Poincar' model of the hyperbolic plane is the interior of the unit circle endowed with the metric ds
( 1)
2
=
4
dX~
+
dX~
which has curvature minus one. The isometries in this geometry are those Moebius transformations that map
2
2
< 1 onto itself. 1 2 An isometry is oompletely determined by the requirement that it x
+ x
oarry a given line-element (such elements are
understoo~'to
be di-
reoted) into another such element. Of course, isometries leave all quantities unohanged which are determined by ds only., angles (= euclidean angles), element of area dA
(2 )
Geodesics are carried into geodesios. They are the
al"OS
2 2 in xl +)(2 < 1
of the cirQles orthogonal to the unit oirole. Hyperbolic distanoe between two points x = (x' ,x ), x' = (x',x') is denoted by s(x,x'). 12 1 2 Consider now thethreedimensional space of line~elements e in the hyperbolic plane. Each isometry of this plane induces, of course, a mapping of that spaoe onto itself. We introduoe as metric in the
line~element
dowhere
d~
2
2
= ds'
spaoe the expression + dX
2
is determined in the following way. Consider two nearby
elements e, e' with bearer points x, x'. Move e from x to x' by parallel displacement, in other words, move the element to x' along the geodesic from x to x' in such a way that its direction makes always the same angle with the geodesic.
d~
is then the an-
gle between the element e in its final position and e'. Obviously,
91
- 44 -
E.Hopf
Id X. I
is independent of the order of the two elements. dO" is there-
fore a Riemannian metrio in e-space. dO" is even invariant under the mappings induced by the isometries of the plane since
the operation
of parallel displacement .is .invariant underisometries. In other words, those mappings are themselves .isometries in e_space relative to dO" . We denote .by O"(e,e') the invariant distance defined by dO" in e-space. The invariant element of
volum~-meaeure
induoed by
dO" in e-space is found to be (4)
dm = d¢dA
where d¢ is the angle-differential in a point of the plane. If we consider in
e~space
the motion along the geodesios with
speed ds/dt = 1 we obtain a one-parameter group of mappings Tte, t +s t s t = T T , of e-spaoe onto itself: T e is the position attained T by e after it has moved t units along the geodesic determined by it. This is the geodetic flow in the e-space of the hyperbolic plane. It is a well-known faot of differential geometry that the geodetio flow on a surfaoe leaves the element of measure (4) in
e~spa-
oe invariant. The same is, of course, true about the Lebesgue measure m determined by it in e:space. m is invariant not only under isometries but also under the geodetic flow. We need the following simple ooordinate-representation of the geodesics and the flow along
t~em.
We assume throughout that
I
"dro-
desic" is a direoted geodesio. A geodesic is oharaoterized by Its points of infinity
e , e+ on
an angle (mod 2n). And a
the unit circle, ~
line~element
e
~s
~
e+,
whero
8
is
characterized by the geo-
desic determined by it and by the position of its bearer point on this geodesic
(orthog~nalcirdular
characterized by the distance
>
arc) .. This p08ition is, in turn,
.s =·0 ,of the point from the eucli.de"n
1 occurs among them. A closed surface y is of first cl 'lsS ;',"' 3 (the sum of the three angles must be < rr) and take
ir :~
t~9 t t r e!
hype r bolio refleotions at the three sides as generators of a ir nup G . The oondition on the angle insures that the images under G of the triangular domain oover the hyperbolio plane simply or, i n
0-
ther words, that this triangle is actually a fundamental domain for this group G . The surfaoe y generated by G is then this tr i angular area, and the geodesios problem is the hyperbolio billiard problem. Quite generally, we oan say that a surfaoe y is of first olass if its area A is finite. The reason is this. If y has finite area its P-spaoe fl has finite volume m(fl)
= 2rrA.
By virtue of Poinoa r/
:3
reourrenoe theorem, almost all streamlines of the geodesio flow in
Q (with .invariant measure m) are reourrent. However, if y were of second olass then, by virtue of the theorem of the next paragraph, almost every streamline would be divergent whioh is a oontradiotion. There are many types of surfaoes y of finite area but of infinit e non-euolidean diameter. A well-known example is that of the
mod ul ~c
group G whioh has a geodetio triangle for fundamental domain w~t h 2 2 one oorner or, rather, cusp at infinity (on Xl + x 2 = 1). Tb.", cusp
99
- 52 E.Hopf does not prevent the domain from having a finite area. A surfaoe y of seoond olass is realized if the fundamental domain D of the group G has on its boundary an open aro of the unit oirole. Any geodetio ray that ends up on this aro must be divergen t on y . In faot, suoh a ray stays ultimately in D and the point ru n ning along it gets farther and farther removed, in the sense of the distanoe s, from the finite part of the boundary of D. .
0
its minimal distanoe from all the pOlnts Sx , S to a fixed point x
o
~
Conse~uently )
identity, congru ent
inside D (they are outside of D) tends a fortio-
ri to infinity. Obviously, the initial directions of the geodetic rays issuing from a fixed point and ending on that aro fill a whole angle. To the arc cn the unit oircle there corresponds an infinite funn.l cf the surface y generated (at least this is true if y has no finite boundary points). Hcwever, surfaoes of second olass oan be much more oomplicated in that they can reaoh to infinity in muoh more intrioate ways. We mention that the olassioal two "kinds" of surfaoes are not the same as our two "olasses". "kind" is a purely topological and, as such, extremely natural motion. From the standpoint of ergodio theory, however, the natural division is the one into the two "c15sses". A more detailed discussion of the relation between "kinds· and "classes" can be found in our memoir [11].
2d. ERGODIC THEORY AND THE TWO CLASSES OF SURFACES Y We are now ready to begin with the proof of the two principal theorems on the geodetio flow on surfaces y . FIRST THEOREM. For a surfaoe y of first class the geode t ic flow is ergodic. In other words, i f f(P) and g(P) > 0 are m-integrable in
n then 100
- 53 E.Hopf
fOf dm
lim "t'-oo
=
holds for almost every P E 0 in the sense of the measure m. The same holds for the limit as
T
~
-00
•
SECOND THEOREM. For a surfaoe y of second class the geodetio flow is dissipative or} in other words} for almost every P E 0 in the t
sense of the measure m the streamline T P is divergent in 0 } posi' tively as well as negatively. Both these theorems have a common root} namely} the existenoe of asymptotic geodesics and the relations between them peculiar to hyperbolic geometry. Their effect is expressed in the following. PRINCIPAL LEMMA. Let B+ and B
be two m-measurable sets in the line-
element space 0 of a surface y . Suppose that they satisfy the following conditions. a) Each set is invariant under the geodetio flow . b) With every streamline in B+ every .streamline pos. asymptotic to it is in B+i the same holds for B
with respect to neg. asymptoti-
oity. c) The set of all points P in one set but not in the other has measure m
= O.
Then} under these conditions} either B+} B
have both measure m re m
=0
or their complements in 0 have both measu-
= O. First we prove this lemma and then the two main theorems. It
is interesting to observe that the lemma is true if in the hypothesis c) and in the oonolusion the statement "set of measure m
= 0"
is replaoed by the statement "empty set". In this altered form the lemma is not only true but almost trivial. In faot} 0) means then that B+ B+
=B
=B =n
} and the oonclusion says that either B+
= B_ = 0
or
This modified lemma is true simply because to two ar-
bitrary geodesics on y there always exists at least one geodesio on y whioh is neg. asymptotic to the first and pos. asymptotio to
101
- 54 E.Hopf the seoond (on the oqvering surface there is preoisely one suoh
=B
geodesio). If Bt
is non-void
this argument oan be applied
the~
t
to an entirely arbitrary streaml.ine and to a streamline in B . However, we need the lemma in the form stated previously beoause, in hoth our applioations, the exceptional nUlsets in the hypothesis c) are, in general, non-void. The proof of the principal lemma rests upon the same simple gument but then the exceptional sets of measure zero
re~uire
~r-
care-
ful handling. A set of complete streamlines in the element-space
n of
a surface y can be represented in two ways, as a flow- inv ari atL
point set in
n
e+)
and as set of pairs of points (8 ,
on the un::t
circle which is invariant under the simultaneous transformations
E G . Remember that t ffde de is a measure
S
n has
n.
= 0
t
measure m = 0
only i f the corresponding set of pairs of points
ffde-Jet
_
in the space of those pairs
ve first: A flow-invariant point set in
sure
n and (e , e ). We
m is the flow-invariant measure in
(e , e ) +
that pro-
i f and
has mea-
To see this -denote by 0 the invariant set in
Let 0' be the set of all elements e in the covering surface
2
2
Xl + x 2 < 1 which represent the elements contained in O. We know that m(O) = 0 implies m(O') = 0 and vice versa. 0' is invariant under the geodetic flow in the covering surface. On using the coordinates
(e-- , e+,
s) in (6) for the elements of 0' we see that 0' is
a cylindrical set, with its base in
-
t
(e , e ) -space
e~ual
to th ft
set of pairs of points. By virtue of ( 7 ) , m(0' ) = o means that that - + set has melisure de = 0 As P is > 0 and continuous this - + The statement is thede = 0 means that that set has -measure
ffpde
ffde
reby proved. The prinoipal lemma is now proved as follows. We assume that
102
- 55 E.Hopf m(B ) > 0 and show that B
=
n-
and B+ =
B
n
-
B+ have measure m = 0
Re-
taining the letters B , B+ for the sets oorresponding to these sets + in (e , e )-spaoe we infer from the hypothesis b) of the lemma that
-
both those sets are produot sets in that spaoe ~
(10)
B
=b
X
Q,
B+
=Q
X b+
where Q is the e-line and where b_, b+ are oertain measurable subsets of it. Striotly, we would have to exempt the diagonal-line
e
=
e+
GX G
from
sinoe no geodesios oorrespond to the points
on' this' line. Sinoe, however, it is of measure
IIde +de +
0 it may
safely be disregarded in this proof. Hypothesis 0) says that B • B+ = b X b+
(11 )
have measure zero. This may be understood in the sense of the produot measure
IIde-de+.
By virtue of (9), b_ has non-zero measure
on the e-line. Consequently, sinoe the first set has produot measure zero, b
+
has e-measure zero. Sinoe the seoond set has produot
measure zero it now follows that b
has e-measure zero. In
other
words, the oomplements of botli sets (10) have measure zero. These
-
oomplements in produotrspaoe oorrespond exaotly to the sets B_, B+ in the spaoe m
= O.
n.
Consequently, these latter sets have measure
The lemma is herewith oompletely proved.
We prove the seoond theorem first. Denote by B_(B+) the set of all elements P 6
n
on neg. (pos.) divergent streamlines . As y
is of seoond olass the angular measure
Ide
of the "pos. divergent"
direotions in a point p € Y is positive. Henoe
JJ fit
d¢dA =
I
y
{fd¢}dA > 0 .
103
- 56 E.Hopf That the two sets Band B+ satisfy hypothesis c) of the principal lemma follows from a general theorem ([g] or [10]) If Ttp, TOp = P, Tt +s = TtT B, is a continuous flow in a comple-
te metric space
n with
invariant measure m
(m u-finite) then the
set of all streamlines that are divergent in one direction but not the other has measure m
= O.
That hypothesis b) is satisfied was remarked before. Therefore, the conclusion of the principal lemma holds, and the second thearem is thereby proved. By virtue of lemma 3 we have obtained a sharper result: If Y is of second class then, in any point p
~
y, the geodesic
rays issuing from p are divergent for almost all initial directions . We now prove the first theorem. Just as in the beginning of the preceding proof we infer that, for a surface y of first class, the set of all pos. or neg. divergent streamlines in sure m
= O.
n is
a set of mea-
We have to mention now that the general theorem
refer-
red to in the preceding proof is part of the following general theorem ([g] or [10]) : Under the same hypothesis as in that theorem
n splits
into
two invariant parts, the conservative and the dissipative part. The first contains almost no pos. or neg. divergent streamlines. The second consists almost excluBively of pos. as well as neg. divergent streamlines. In the conservative part 00
(12 )
J
o
t
geT P)dt
=
00
holds for any g(P) > 0 almost everywhere. In the first chapter of these lectures this decomposition was mentioned already for the case of a single mapping T (however, without the part about divergent streamlines).
104
-
~7
E.Hopf
In our present case, geodetic flow on a surface y of first class, the flow is purely conservative. For a conservative flow with invariant measure m the ergodic theorem states this [10) : If f(P), g(P) > 0 belong to L (m-integrable) in 11 then m 7 f o f(Ttp)dt lim q (P) = f 'It' (P) , (13) q (p) = 7
7: ....00
7
f7 g(Ttp)dt o
exists almost everywhere (m) in 11
gf If €
Lm ' f ll'
t
is T -invariant
and satisfies the relation
f
(14)
11
=f
gf*hdm
- 11
fhdm t
for every bounded and measurable h(P) that is invariant under T . The average in the past, lim
(15)
q (P)
7 .... -00
= f /tit
(p)
7
"'t -t exists almost everywhere too (apply theorem to the flow T = T
Tt), and · which has the same invariant f unc t lons as the same .relation (14) as
f
11
g (f
f~~ sa t"lS f"les
f~. Consequently,
** - f'f) dm
= 0
must hold for every bounded invariant h, so for instance, for h
= sign
(f** - f*). Hence, f
(16)
*"*" (P) = f~
a. e.
(p)
must hold. Our aim is to show that f h
= 1,
*" =
fW-*
is constant a. e .. From (14)
it would then follow that this oonstant has the value ff/fg ·
Tc prove constancy of f *for any f
'L m
it suffices to prove this
for every f in a set of f's that is dense in Lm' Reason: The linear operator fit" = T*'f (g' is kept fixed) satisfies
105
- 58 .. E.Hopf
J
n
* I dm .s f nI f Idm
g If
(apply (14) with h = sign (f~ )}. We use this fact as fJllows in
n).
our present oase (y,
We choose the fixed function g > 0 such
that g(P') - g(P) (17)
holds
uniform~y
a(P,P') - 0
as
g (P' )
with respect to P, P'. In the important speoial ca ..
se where y has finite area we may simply choose g
= 1.
For funct io ns
f we take only those f's which have the similar property that (18)
holds
fep')
~
f(P)
o
g (P' ) uniform~y
in
n
a(P,P') ... 0
as
In the case where y has finite area and whe-
re g = 1 this simply means that f is uniformly continuous in
n
It can be shown by means of claSsioal arguments that the set of those f's is dense in L . m After these preparations we now turn to be main point of our proof. Consider g, f, as indicated above. We show first that ( (19 )
f(Ttp')dt
ro g(Ttp')dt
have the same limits as
if the two streamlines occurring he-
T ~ 00
ft' g , f', g' t t t integrands, respectively. Then by virtue of lemma 2 and in consequenre are pos. asymptotio. Write briefly
ce of (17) and (18), the quotients at =
tend to zero as t
~~.
l' '
t
-
-
f 't
bt
g' t
=
g' t
-
gt
g' t
The differenoe of the two quotients (19)
may be written as
106
- 59 E.Hopf T
I
T
0
T
Io
atgtdt,
t
Iog'dt t T
Iobtg'dt t
f dt t
t
g dt t
0
g'dT 0
T
This equality and (12) show that the difference tends to zero as 1)
Obviously, the same is true as
if the two stream-
T ~ -00
lines are neg. asymptotic. Consequently, the two invari,lllt sets
=
Bt
[p If*' (P) .? c]
satisfy hypothesis b) of the principal lemma no matter what the value of the constant cis.
(16) implies the validity of hypothes i s
c) . From that lemma we can, therefore, infer that either Bt or its complement has measure m in other words, f~
=0
. This holds whatever value c
h~s
o~,
is constant a.e., the first theorem is thereby
completely proved. Lemma 3 permits again to state the theorem with a
r
clusion but a stronger hypothesis: If
sh~rper
con-
is of first class, if f(P),
g(P) > 0 are in Lm and if these functions have the continuity properties (17) and (18) then lim qT(P) point p f
r
= IfdmlIgdm
holds in every
for almost ; all directions (P) in p.
2e. ADDITIONAL RESULTS AND PROBLEMS. The previous results have
str~ightforw~rd
complete n-dimensional manifolds
l7t of
generalizations to
scalar curvature minus on e .
The universal oovering space is the interior of the unit sphere 2
2 x
1)
1
tX
2
2 t".tX
n
t ex aotly the same ergodio theory remains vali.d if 1 is a oomplete two-dimensional Riemannian .manifold of variable negative curvature (1 is sumed to be of differentiability olass
C'~.
88-
There is the same basic
diohotomy into two olasses, and both first and seoond theorem remain literally valid provided that the ourvature remains .between negative bounds. The mixture property has, however, not yet been proved for variable curvature. We would like to return onoe more to our surfaoes 1 of
' oon~
stant negative ourvature. We mentioned above that the purely topologioal subdivision into two kinds is different from our subdivision into
two~lasses .
The first kind is aotually a larger totali-
ty than the first olass. Take a surfaoe 1 of first kind but of seoond olass. Paul Koebe had already proved, in his famous memoirs on
non~euolidean spaoe~forms,
that on any '1 of first kind there e-
xists a quasi-ergodio geodesio (streamline dense in D).l) For a general flow that .isergodio (f* = oonst. for every f) the same thing is true; even "most" streamlines are quasi_ergodio. What 1)-------~--·
-
Koebe's topologioal analysis of the geodetioal flow on surfaoes 1 has been oarriedo'n, and several of his results sharpened, by Gottsohalk and Hedlund [51. There is the following result on"topologioal" mixture on surf~oes y of the first "kind" (this is sher~ar thari Koebe's rrsu1t mentioned above): If A is a non-void open subset of n then. TA 'oeoomesdenserand denser inD as t -> ro. To Hedlund [8] the ergodio theory of the geodesios on surfaoes 1 owes the first (and an ingeneous proof it is) proof of the JIleasure-theoretioal,mixture of the flow on a surfaoe1 'of finite area. The author~s proof of the same faot for n_dimensional manifolds 17t(quoted above) would not have been possibLe without Hedlund's beautiful idea. The author's .first proof ,of ergodio mixture for surfaoes 1 of finite area [13] had yielded mixture only in a less stringent sense.
109
- 62 E.Hopf ~bout
the oonverse question: Does the existenoe of a
qU~Biergod10
streamline imply ergodioity of the flow? The answer is "no". On a surfaoe y of first kind but seoond olass the geodetio flow is' d1s· sipative .
110
E.Hopf REF ERE NeE S
1.
C.CARATHEODORY: Bemerkungen zum Ergodensatz von G.Birkhoff. Sitzungsber. B"Iyr. Akad. Wiss", Math. -Naturwiss. Abt. 1944(1947) p.189-208.
2.
R.V.CHACON and D.S.ORNSTEIN: A .gener"ll ergodic theorem. Illinois J. of M:lth. 4 (1960), p.153-160.
3.
J.L.DOOB: Asymptotic properties of Markoff transition probabi1 i tie s. T r:l n s . Am e r. Mat h . Soc. 63 (1948), p. 393 - 4 21
4.
Y. N. DOWKER: A new probf of the. general ergodic theorem. Act'l. Sci. Math: Szeged 12 (1950) p.162-166.
,5.
W.H.GOTTSCHALK 'l.nd G.A.HEDLUND:Topological dynamics . Amer. M.' l.th. Soc. Colloquium Publicati.ons 36 (1955).
6.
P. R.HALMOS: An ergodic theorem. Proc. N'>t. Ac,>d. Sci. p . 156-161.
7.
P.R.HALMOS: Measure theory. New York 1958.
8.
G,A.HEDLUND: The dyn!l.mics of geodesic flows. Soo. 45 (19~9), p.241-260.
Bull.
32 (1946)
Amer. M:lth.
" Zwei Satze uber den VerI auf der Bewegungen dynamischer Systeme. Math . Annalen 103 (1930), p . 710-719.
E HOPF:
10.
EL HOPF:
11.
" E. HOPF: St:ltisti* der geod'ltischen Linien in ,Mannigfal tigkeiten negativer Krummung. , Beriohte S~ohs. Ak'l.d. Wiss." Math.N9.turwias. Klasse 91 (1939), p.261-304.
12.
Ergodentheorie. Berlin 1937.
" II E.HOPF: St9.tistik der Losungen geod'l.tischer Prob.leme vom unst9.bilen Typus II. Math. Ann'l.len 117 (1940), p.590-608.
E. HOPF: Beweis der Mischungscharakters der geod~tischen Str~ mung auf vollst~ndigen FlBehen der Krilmmung minus Eins und ~u dlioher ·OberflMche. Sitzungsber. Preuss. Akad. Wiss." Phys.math. Klasse 30 (1938).
14.
E.HOPF: The general temporally discrete Markoff process. J. R:ct Meeh. An'l.l. 3 (1954)., p. 13-45.
15·
E.HOPF: On the ergodic theorem for positive linear operators . J , reine und angew. Math. 205 (1960), p.101-106.
16.
W. HUREWICZ: Ergodic .. theorem without invuiant me'l.sure. 45 (1944), p. 192-206.
17.
S.KAKUTANI: Ergodic theorems and the M'l.rkoff process with a stable distribution. Proe. Imper . Acad. Tokyo 16 (1940); p.49-54.
111
Ann. Math
- 64 E.Hopf 18.
S.KAKUTANI: Ergodic theory. Proc. Int. Congress Math. Cubridge, Mass., Vol. 2 (1952), p.128-142.
19.
J.C.OXTOBY: On the ergodic theorem of Hurewicz. Ann. Math. 49 (1948), p.872-884.
20.
K.YOSHIDA and S.KAKUTANI: Operator-theoretioal treatment of Markoff's prooess and mean ergodic theorem. Ann. Math. 42 (1941 ) , p.188-228.
112
CENTRO INTERNA'ZIONALE MATEMATICO ESTIVO (C.I.M.E.)
J 0 S E'
MAS S ERA
LES EQUATIONS DIFFERENTIELLES LINEAIRES DANS LES ESPACES DE BANACH
ROMA - Istituto Matematico dell'Universitl - 1960 113
- 1 -
J.Massera INTRODUCTION Nous nous proposons de faire un expose sommaire du oontenu de p1usieurs Memoires eorits en oollaboration aveoJ~J~Sch~ffer et qui ont ete pub1ies sous 1e titre general Linear differentia~ equations
and functionaL anaLysis, Parties I, II, eto. [2, 3, 4, 5, 6, 7, 10, 12]; 1es theoremes, formu1es, etc. de ces Memoires seront oites
P!H
1a suite de 1a fa90n suivante: Theorhe II.5.4, formu1e 1.(2.1) , etc. II nous sera aussi necessaire de resumer les principaux resu11\
tats de J.J.Sohaffer dans son Memoire Function ipaces with
transLa~
tions [11], que nous oiterons F.3.2,F.(4.1), eto. L'obje .t d I etude . sont les equations
ou
t €
x
+ A(t lx = 0
(1)
x
+ A(t lx = f(t)
(2 )
x
+ A(t lx = h(x,t)
0)
= [0,(0); x, f, h
J
A(t) ) pour chaque
t
E X , un espaoe de Banaoh queloonque ;
fixe, est un endomorphisme (continu) de X ;
A(t), f(t) sont desfonotions integrables (Boohner, au sens cbnven a.ble) dans ohaque sous-intervalle borne JI C J . Dans les annees 1930-35,
o. Perron [aJ,
K.P. Persidskii [9] et
r . G.Malkin [1] etablirent· l'equivalenoe des p~oprietes suivantes (dans Ie cas dim X < (p 1) Pour ohaque
00 ,
A(t) oontinue):
f
oontinue ' bornee dans
J (11 espace for me
par ces functions, avec la norme . duo supre/llum, sera designe dans ce qui suit par ~), toutes lea soluti~ns de (2) lerons des
€
t
(nous les appe-
C -solutions).
(p ) Pour ohaque 2
h
oontinue, Ilhll ~ (3,' IIh(x',t) -h(x",t ) ll..::
.:s A Ilx'
- x" II, aveo (3,A auffisamment petites, toutes 1es s o l ut i ons
de (3)
t En
r~alit',
0
+ T(€)
0 Jet
II x(t) II
.
Perron a ddmontrd l'dquivalence des propridtds
suivantes, plus gdnerale que l'equiva1enoe de (p 1 ) et (P 2 ) : (Ql) Pour ohaque f € (0 ~ k ~ dim X) de
(Q2) Sous les metres de
t
il y a une famille • k parametres
~-solutions de (2). hypoth~ses
e-solutions
Dans nos travaux
de (P 2 ) i1 Y a une famille • k para-
de (3).
I~III
nous nous avons posd la question de
formuler adequatement des proprHtes analogues de hOon a rdtablir l'd,qu:uvoahn!oe 6lv,ed , ~ ) est a,d1llissib~e, d te une consta,nteK >0 te~ Le que pour cha,que f
f'f.>
exis .-
i ~ y a, une e t u-
ne seuLe so~ution x(t) de (2) sa,tisfa,iSllnt x E'iJ , x(O) E X 1X), et on a,
I xI~
~
d d~
La demonstration de oe r'sultat repose sur Ie
theor~me
de gra-
phe ferm' suivant qui peut se · deduire des inegalites (5) et f6) :
123
- 10 -
J. Ma ssera
t
THEOREME 2 (Lemme IV.2.1). Soit{f}C n
(X) et x
n
une soLution de
L
i + A(t)x = f n . Si .{f n } et .{x n } convergent dans respectivement ·versf } x , on a x + A(t)x = f et x -- x uniformernent ·dans chaque n
sous-intervaLLe borne.
DICHOTOMIES La generalisation naturelle de la propriete (P 4 ) est Ie type de oomportement des solutions de (1) que nous avons appele di c hotomie 8 (simples ou exponentielles) et que, comme nous verrons en s u i te , est etroitement lie
a
l'admissibilite de oertains couples pour l'e-
quation (2). En vue de (P 5 ) on peut aussi designer oes types de com portement par stabiLite conditionneLLe uniforme (simple ou asympt otique) de la solution x
=0
de (1); Ie oas asymptotique est essentiel-
lement equivalent, pour les equations lineaires, a oe que N. N.Kr asovs kii appelle comportement non-critique uniforme (of.
Theor ~ me
I .3. 5,
Corollaire I.3.1 et Exemples I.3.3 et I.3.4). ( 1)
Nous dirons que les sous-espaoes oomplementaires Y J Y de X o 1 induisent une dichotomie expon~ntieLLe des solutions de (1) s'il existe des oonstantes positives N , N' , oonditions suivantes sont
V
,
v' , y
o
telles que le s
rempli~s
(Eil Pour ohaque solution x(t) de (1) _aveo x(0) € Y Ilx(t) II
~
Ne ~V(t-to7 IIx(t ) II, t o .
~
t
0
~
on a
0
o;
(Eii) Pour ohaque solution x( t) de (1) avec x (0) v' (t-t ) IIx(t) II ~ N'e 0 IIx(t ) II , t ~ to ~ 0 ;
~
Y
1
on a
0
(Eiii) Pour ohaque oouple de solutions x. (t) de (1) avec ~
~ y , t ~ 0 . 010 Si dans (Ei), (Eii) on supprime les faoteurs exponer:tiel s, on
x (0) £ Y. , i = 0, 1, on a
y. (x (t), x (t))
obtient les proprietes (DU,
(DiU, lesquelles, avec (Diii ) = (Eiii),
i
(1)
1
On peut formuler la definition de fa90n a ne pas fai r e i nte rveni, Y1
124
-
11 -
J.Massera ddfinissent Ie oomportement appeld dichotomie (simp'e) des solutions de (1) indui te par les sous-espaoes Y ,Y o
1
.
On peut demontrer (Lemme . 1.5.4) que si A f
M alors (Eil et (Eii)
impliquent (Eiii). Dans Ie oas Ie plus simple, on a : LEMME 1. 81 dim X · (00
,
A = const. et si 'es pa.rties reeHes des
ra.cines ca.ra.cteristiques de -A ·sont
non~lIuLLes,
on a. une dichotomie
exponentieLLe; et reciproquement.Si a.ux ra.cines ca.ra.cteristiques purement ima.gina.ires correspondent ·des diviseurs eLementa.ires Linea.ires, a.Lors on a. une (etPeut~6trePLusieurs) dichotomie simpLe; et reciproquement. On peut aussi envisager desoomportements "en moyenne" ou "par tranohes" du type suivant : On a la sta.biLite a.symptotique uniforme en moyenne s'il existent des sous-espaoes oomplementaires Yo ' Y1 ' des oonstantes positives v , v' et des fonotions positives M(tl), M'(tl) ne dependent que de tl > 0 , tels que : (Mi) Pour ohaque solution x(t) de (1) avec x(O) (Yon a o
t ttl
J
t
Ilx(u) II du
(
)
t ttl
~ M(tl) e -v t-t o J
0
to
Ilx(u) II du
t
~
t
> 0
o
(Mii) Pour ohaque solution x(t) de (1) avec x(O) f Y. on a 1
t+tl
J
t
Ilx(u) Iidu ~ M' (tl)e
On a la vement
sta.bi~ite
v'(t-t ) to+tl 0
a.symptotique
J
to
Ilx(u}lIdu,
~niforme
t
~ t
o
~
0 .
pa.r tra.nches (relati-
a un espaoe fonotionnel donne lJ · oontenant les fonotions
oaraoteristiques des sous-intervalles bornes) s'il existent Y
o
Y1 ' v , v' , y(tl) , M' (tl) comme plus haut tels que :
qui peut meme ne pas exister comme complement de Y Yo est ferme est, par oontre, essentiel. 0
125
Ie fait que
- 12 J.Massera (Til Pour chaque solution x(t) de (1) avec x(O)
I x[ t,t+AJ
x' ~
-v(t-t ) 0
M(6)e
)j
I t rto,to+AJ xl, X)
€ Y
on a
o
t~t
~O,
o
~
(ou
designe la fonction oare.oteristique de 1 I ensemble E C J); E (Tii) Pour ohe.que solution x(t) de (1) aveo x(O) E Y l on a
t
[t,t+AJ
x,
~
~M'(A)e
v'(t-t)
01%
[t ,t +A] o 0
xl,
t~t
X)
o
~O
.
8i l'on supprime les fe.oteurs exponentiels, on obtient les oomportements simp~es respeotifs.
II n'y a pas apparemment de oorre-
sponde.nt naturel de la propriete (Eiii) dans les oas envisages maintenant.
THEOREME8 FONDAMENTAUX On peut demontrer les theorbmes suivants THEOREME
3 (Theoreme!) IV.6.1 et Iv.6.2). Si (~,n) est un
cQuPbe admissibLe d'espaces decLasse
r
r
(un
-coupLe), iL y a
stabiLitt uniformeen mo,enne.etpar tranches des soLutions de (1) Yo
= Xo~ , Yl = Xl,.:}' Si (tJ, 1)
, Go ) ·00
n' est pas pLus faib Le que
La stabiLitt est mIme asymptotique.
THEOREME4(Theoreme
admissibLe PLus fort que (
IV.S.l). Si
t1
e
,
OO
,X
)
(1D,V) 02J
,X
est un
lt)
T-coupLe
induisent une dicho-
tomie des soLutions de (1). THEOREME
admissib Le et
5 (Theoreme A
e: 7ft ,
rv.S.2).Si
(!b,iJ) est un
T-coupLe
Xod)' Xl~ induisent une dichotomie des so Lu-
tionsde (1). THEOREME
tions de (1),
p~es
6
(Theoreme IV.S.3).
Le coupLe
(f} ,
eoo )
PLus faibLe$)j si dim Y
~ (Example 11.2.1).
THEOREME 16 (Corollaire 11.3.3).Soit A(t) periodique de periode 1 at supposons
que La farmeture ,de 'Xo'e
so it refLexive. Etant donne
une fperiodique de peri ode 1 I, si L'.equation (2)admet une soLutionbornee, e,LLe admetunesoLution periodiqus Ide ·periode 1 THEOREME 17 (Ths9remeII.3.5). 8'iL exists un T-coupLe admis-
sibLe.qui n'est pas Hus faiHe que
('1.\
e:)
et si A(t), f(t) sont
.pedodiques de p.edode 1 , L'equation (2) admet une et une seuLe Lution
p~riodique
50 -
de p4riode 1.
f) Casou A est oonstant (of. Lemme 1 et
[10]) ;
THEOREME 18. Si A = canst., pour qu'iLy ait dichotomie exponen-
.tieL'e des soLutions de (1) iL faut et iL suffit que Le spectre de A
n. coupe ·pas L'axe imaginaire.
130
- 17 J.Massera
BIB L I 1. LG.MALKIN,
a
GR A PHI E
On stabiLity in the first .a;pno:dmation, Sbornik
Nauonyh Trudov Kazanskogo Aviaoionnogo Instituta, 3 (1935),
7-17
(en russel.
,.
2. J.L.MASSERA.and J.J.SCHAFFER, Linear differentiaL equations and
junctionaL ,anaLysis, I, Annals of Math., 67 (1958L 517-573.
3.
J.L.MASSERA andJ.J.SCHXFFER, Li~ear differentiaL equations .and
junctionaL ,anaLysis,II. Bquations ,with ·periodic coefficients, ibid, 69 (1959>, 88-104. 4. J.L.MASSERA and J.J.SCH~FFER, Linear differentiaL equations and
functionaL anaLysis, III. Lyapunov's second .method in the case of condicionaL stabiLity, ibid, 69 (1959), 535-574.
5.
J.L.MASSERA and J.J.SCHXFFER, Linear ,differentiaL equations and
junctionaL anaLysis, IV, Math.Annalen, 139 (1960),287-342. 6. J.L.MASSERA, Un criterio de e%istenciade soLuciones casi-perio-
dicas de ciertos sistemas de ecuaciones .diferenciaLes casj-periodicas, Publ.lnst.,Mat.Estad. (Montevideo}, III (1958), 99-103.
7
J.L.MASSERA,
des Mat.
1ur
L~e%istence
de.soLutions bornies et piriodiques
syst~mes . quasi~Ljn.aires , d~iquations
diff.rentieL.es, Annali di
(IV) 51 (1960), 95-106.
8. O.PERRON, Die StabHit8tsfrage bei DifferentiaLgLeichungen, 'Math. Zeitschrift, 32 (1930}, 703-728.
9. K.P.PERSIDSKII, On the stabiLity of motion in the first a,pproxi-
ma,tion, Mat.Sbornik, (N.S.) 40 (1933}, 284-293 (en russel. 10. J.J.SCH~FFER, Ecuaciones diferenciaLes LineaLes con coeficientes constantes en espados de Banach, Pub1. Inst. Mat. Estad. (Montevideo), III (1958), 105-110.
131
- 18 -
J.Ma sser a II
11. J.J.SCHAFFER, Function
sp~ces
with transLations, Math. Ann.le n,
137 (1959}, 209-262; .4ddndulr J ibid, 138 (l959}., 141-144. 12. J.J.SCH~FFER, Ljne~r Dffjerenti~L equatjons ~nd
naLysis, V, Math.Annalen, 140 (1960),3 0 8-321.
132
functionaL a-
CENTRO INTERNAZIONALE NATEMATICO ESTIVO 'C.J.·M"E. )
LUIGI AMERIO
FUNZIONI QUASI .. IIERIODICHE AS'llRA'l'TE E PROBLEM I DI PROPAGAZ]ONE
ROMA - Istituto Matematico dell' Universita :- 1960 133
FUNZIONI QUASI-PERIO VI CHE
ASTR' ? T ~
E PROBLEMI
DIPROPAG AZIONE di L.AMERIO
1.- Esporremo alcuni recenti risultati sulle soluzioni quasi-periodiche (q. p.) dell' equazione delle onde, e, piu in generals, tie ll,} equazioni
differenz i~ l i
Rioordiamo, (1. 1 )
y
inn~zi
astratte.
CJ . p.
.tutto, che una junzione continua
= f(t) ,
t E J
= (-
a valori in uno spazio B , di Banach,
gni
10
e
OJ
+
OJ)
,
quasi~periodica
se ad
0-
>·0 pub jarsi corrispondere un insiellle T ·, rdativalllente . 10
denso, di numeri r (~uasi-,pHiodi) per ciascuno ,dei . quaLi r;isuLti (1. 2)
Sup
lint + r ) - f(t)
II
~
10
•
Q1,lesta definizione si riduce, se B
e
t EJ
euolideo, .a quella,
classical di Bohr. Vale .inoltre, . anche nel caso astratto, il fon. '. " . e su ff'1dam ent.le crlterlo dl Boohner (1) : con d Iztone necessar;za
dente perche f(t), continua inJ, sia ".p.
e
cheda ogni suc-
cessione reaLe .{h n } possa estrarsi una sottosuccessione{h'n } taLe che La successione{f(t -I-h'}} converga unijormemente In J n
Per quanta si dir~ nei §§2 e
3, i nteressa supporre B hiL2
1
bertiano: preoisa.mente .interessano .gli spazi L , HO' E di cui
0-
ra rioorderemo le definizioni. 2 I) L e 10 spazio delle funzioni reali y = y(x) a quadra t o integrabile in un insieme
n (aperto,
spazio euolideo Xm (x = xl"'"
x m),
di prodotto seal are :
135
Limitato e connesso ) dello con la consueta definizione
- 2 -
L,Amerio (y (x), z (x))
(1. 3)
=
L2
f
0
y (x) z (x) dO ,
II) Hlo e 10 spazio delle funzioni reali y = y(x) a quadrato integrabile in 0 insieme aIle derivate prime (nel sense della teoria deLLe distribuzioni) e nulle (in senso generaLizzato) sul190 frontiera
U
di 0 ,
1 Assumeremo oome prodotto soalare, in H , 190 quantita 0
(1. 4)
(y(x), z(x) 1 = H 0
f
l--m oy oz { L + a(x)y(x)z(x)}dO a (x) Ox 'Ox k jk OJ, k j
---
Nella (1.4) Ie ajk(x) = akj~x), a(x) Bono funzioni misurabiLi e 1--m Limitate, e a(x) .? 0 e' La forma quadratica L aJ'k(x) ~J' k
e
j, k
soddisfa aLLa Limitazione a
(x) ~
jkj
gk
m
>
~
/J.
r
2
L1J' 'J'
1
Rioordiamo ohe H· si ottiene compLetando, rispetto alIa norma in-
o
dotta dalla (1,4), 10 spazio delle funzioni oontinue in 0 insieme aIle loro derivate prime, e nulle in un intorno della frontiera U
,
E'
' 1 tre Hl0 C L2 ,
lItO
Ill) E e 10 spazio prodotto oartesiano di H1 per L2
o
E = Hl X L 2
o
Ogni elemento Y(x) E E
e
peroia oostituito da una coppia 1
2
{Yo(x), Yl(x)} di funzioni : Yo(x) E HO' Y1 (x) E L , 8i assumera oome prodotto soalare, in E, 190 quantita, (1. 5)
(Y(x), Z(x))
E
= (y (x), zO(x) 1 + (y1(x), z (x)) 2 '
0
H
o
oui oorrisponde 190 norma (1. 6)
136
1
L
- 3 L.Amerio Chiameremo E 10 spazio deLL'energia e la metrica (1.6) la
metrica deLL'energia. Inf~tti
Ie quantita
1
2
-2 Ily 0 (x) II H1
o
1
1
2
2
rappresentano rispettivamente l'energia potenziaLe,
L'energia ci-
netica e L' energ'ia totaLe di una membrana i1 , col borde fisso, in cui YO(x) designi 10 spostamento del punto x, Y1 (x) la veLocita del punta medesimo.
2.- II
de
(0
(2.1)
misto, secondo Hadamard, per l'equazione deLLe on-
problem~
equazione deLLa membrana vibrante) -/y at 2
=
l--m a oy I ) (a (x) Ox jk j,k ox . k J (t ,
e
-
~(x)y
+ f(x, t)
J, x \7 11) .
stato oggetto di numerose ricerche, anche recent 1,
nostr~
eaposizione
interess~
(2)
.
Per la
considerare Ie .oLuzioni deboLi del-
la (2.1) : ci riferiremo inoltre al problema definito dalle con-
dizioni iniziaLi (2.2)
y(x,O) :.: YO(x)
con YO(x), Y1 (x) funzioni assegnate) e (2.3)
y(x , t)1
dal1~
condizione ai Limiti
=0, (J'
corrispondente alIa vibrazione di una membrana coL bordo fisso. Le soluzioni deboli si associano alIa teoria variazionaLe, anziche differenziaLe, della
membran~
vista, Ie soluzioni deboli ci
vibrante: sotto questa punto di
~ppaiono
137
come Ie vere soluzioni,
- 4 -
L.Amerio risultando, in piu, soluzioni della (2.1) (soluzioni da dirsi for-
til quando soddisfino, unitamente alIa frontiera
IJ
,
a oonvenien-
ti oondizioni di regolarita. Soluzioni deboLi della (2.1) saranno dette, preoisamente, le soluzioni y(x,t) dell"equazione variazionaLe deLLe onde (oon 1"1. quale si traduoe it principio di Hamilton): (2.4)
/3
J{Yt (x,t), ¢ (x,t)) 2 - (y(x,t), ¢(x,t)) 1 + ~ t L HO
+ (-f (x, t) J ¢ (x, t)) 2} d t = 0 . L
Questa deve essere verifioata (comunque si prenda l"intervallo ~ /--1 (3) in oorrispondenza di tutte Le variazioni ¢(x, t),
nuL Le per t =
~
e per t = (3 . In modo preoiso, oeroheremo Ie so-
luzioni del problema misto nella olasse
r
delle funzioni u(x,t)
soddisfaoenti aIle seguenti oondizioni: 1
A)
u(x,t) sia continua, oome funzione di t a valori in HO;
B)
u(x, t) sia derivabi Le, oome funzione di t a valori in L , e
2
la derivata ut(x,t) risulti continua in J . El" ohiaro ohe, se vaLgono Le A) e B),
La funzione U(x,t) =
- {u(x,t), ut(x,t)}, a va Lori in El, ha ivi, come traiettoria,
una Linea continua. Sulla variazione ¢(x,t) si fanno anoora, nell"intervallo
~H (3, Ie ipotesi A), B); si supporra, in piu, ohe sia 11¢(x,~)
II
1 =
11¢(x,/3) II
H
1
=
o .
HO
o
Quanto al termine noto f(x,t), ammetteremo ohe sia a 2
in L , per quasi tutti i t , e ohe La norma
II f (x, t )11 2 L
val~ri
sill, in-
tegrabiLe in ogni intervaLLo Limitato. Infine, per quanto riguard aid at i in i z i al i, sis up po r r a
0
he, nell e (2 . 2 ), s i a
138
- 5 L.Amerio Y 1 (x)
e
L
2
ei08 Y (x) 0) ~ E •
8i pub allora dimostrare (efr. L.AMERIO) Loe. cit. in (2» ehe ~a so~usione Y(x)t) = {y(x)tL y (x)t)} dena (2.4») con La t
condisione
ini8ia~e
esiste in tutto J )
e unica
e dipende con continuita dai dati, in
virtu della formula di maggiorazione (2.5)
La soluzione medesima pub caleolarsi) seguendo 10 schema classieo delle soluzioni elementari) mediante la formula risolutiva un(x)
00
2: . 'Y (t) 1n n
(2. 6)
An
'Y1(t)
00
2: n .:..::..- u (x)} . \
1
I\n
n
Nella (2.6) le un(x) e le eostaoti An rappresentano rispettivamente Ie autosoLusioni e gli
dell'equazione
= A2 (u(x») hex») 2
(u(x») hex»~ 1
(2.7)
autova~ori
H
L
o
ehe deve intendersi soddisfatta per tutte le hex) €
Hi
o
La (2.7) ammette una suceessione {A } di autovalori) per i
n
quali risulta
o
< Ai < A
8e Be hilbertiano, questa equivale ad ammettere che,
b (; B ,
iL prodotto
sc~Lare
(f(t), b)
In ogni caso, si dimostra che condi2ione
ciente perche f(t),
dd.P., sia ".p.
tori'l, in B , risuLti
rdativ~mente
e
B
risuLti ".p.
necessari~
e suffi-
che La corrispondente traiet-
compatta.
Cib premesso, la dimostrazione si svolge nel modo seguente. I') 8i rioonosce, innan2ii tutto, come in I), che se vale la (3.5), la fun2iione Y(x,t) e d.g.p. II') 8i dimostra che, se vale la (3.5), presa ad arbi trio una successione reale a = {a k }, si pub estrarre da questa una sottosuccessione
a'
= {a'} k
tale che la successione {y (x,t +
144
a'» converga dek
- 11 -
L. Amedo bolmente e uniformemente in J a una funzione Z(x,t), d , q.p. e soluzione dell'equazione variazionale delle onde, oon termine noto
= lim
g(x,t) EI
f(x,t + a k' ) .
k-oo
allora Sup
t 0
oostanti reali
ovvgro il sistema del 1 0 ordine x
2
Y = - k x - b Y
= y
Si tratta di un sistema differenziale nel piano reale R2 od anohe sulla sfera S2
= R2
+
00
(mediante l'ordinaria proiezione ste-
reografioa). Se b = 0 si ha l'osoillatore armonioo. Ogni soluzione, qualunque siano le oondizioni iniziali, e periodioa. Si ha oosl una famiglia di ourve ohiuse intorno all'origine del piano. Se b > 0 si ha l'osoillatore oon attrito. Nel oaso b
=0
non vi e stabilita strutturale poiohe un oambia-
mento della b , per quanto piooolo, oambia radioalmente la oonfigurazione della famiglia delle ourve integrali. Inveoe il oaso b > 0 presenta la stabilita strutturale poiohe un oambiamento suffioientemente piooolo di b in b(x,y) > 0 e analogamente per il coeffioiente k
2
da anoora un osoillatore oon attrito. Consideriamo, in generale, un S i
x
1
iii
si~tema
n
= f ( x , .•. ,x)
n
1
con Ie f (x , ... , x ) ~ C
in
8 ,
i
differenziale reale
= 1,2, ... , n
insieme aperto e limi tato dello
Rn euolideo (reale).
151
- 2 -
L.Markus Per i teoremi di esistenza e di unicite per ogni punto di E) passa una curva integrale. Consideriamo la famiglia di tali curve.
e
Allora S e un oampo di vettori in
ed una ourva integrale e una
ourva tangente a tale oampo. Sia ~
10 spazio di Banach ohe si ottiene prendendo tutti i si~
stemi differenziali come S . Preoisamente oampi vettori~li di classe 0 1 in
G
oonsista di tutti i
e sulla frontiera
'&8 (che
supporremo regolare, ad esempio una varieta differenziabile di dimensione n-1). La distanza tra due campi vettoriali S ed S' iii n i :c g (x , ... , x ) sia
II S
- SI
II
;:
max x E 0 esiste un 0 > 0 tale ohe quando per 8 1
fID
allora S ed S' sono €..,omeomorfi,
un'applioazione topologica di 6)
su
e
lis - s' II
< 0
vale a dire esiste
la quale porta la famiglia
delle curve integrali (non parametrizzate) di 8 sulla famiglia di quelle di 8', in dalla sua immagine
modo tale .che la distanza di ogni punto di (3
e
€ .
~
Allora se si oambiano di pooo, Ie f
i
, la configurazione glo-
bale delle curve integrali non muta qualitativamente. II concetto di stabilita strutturale
e
molto interessante dal
punto di vista filosofioo pOiohe un sistema dinamico della fisica non deve cambiare qualitativamente allorohe si produce un piccolo cambiamento nei ooefficienti. Ma l'idea
e
difficile a trattarsi dal
punto di vista matematioo. Se l'insieme (j
e
aperto e limitato in una variete Mn (inveoe
che in Rn) la teoria procede in modo analogo. In quel che segue sa-
152
);
- 3 L.Markus
e =M
, variet~ differenzia-
(38
= ¢ (insieme vuoto) e
n
rl considerato il CaSo pib semplice
bile compatta. In tal CasO la frontiera
la condizione 1. nella def. 6 sempre verificata.
SCHEMA DELLA TEORIA. La teoria dei sistemi strutturalmente stabili si pub. descrivere nei seguenti punti : 1. Esiatenza; 2. Denait?> in
7j ; 3. Proprietl;
4. Applicazioni. Ci soffermeremo in particolare sui due ultimi e accenniamo rapid lmente ai .primi lIIue.
L'esistenza di sistemi strutturalmente stabili e stata provata nel caso di un diaoo da Andronov e Pontriaghin (v. Bibliografia) e 2
la prova ai puo estendere ad ogni M ne di S.Smale .citata
nell~
Quanto alIa den,sita in
3
per M ai veda la comunicazio4
Bibliografia. Coaa puo dirsi per M ?
lb ,
vale a dire al fatto che i sistemi
strutturalmente stabili sono densi nello spazio
~
di tutti i si-
stemi, essa risulta finora provata solo in U2 (vedi De Baggis e Pei3 xoto). Cosa puo dirsi in II ?
PROPRIETA' . Elenchiamo adesso alcune proprieta relative ai sistemi strutn
turalmente stabili in unavarietk differenziabile compatta M . 1. Esiste un numero finito di punti singolari (~unti d'equilibrio) ed ognuno di essi e un punto elementare. (Cio significa che se P e un punto singolare,oioe un punta in cui il vettore del campo
~
nullo, esiste un'applicazione topologioa di un intorno di P n
su di un intorno dell'origine di R
che porta Ie soluzioni di S su
quelle di una equazione lineare a coefficienti costan·ti
ici =
a
i
xj
j
153
- 4 L.Mark~s
tale ohe gli autovalori della matrioe (a i ) hanno tutti parti reali j
diverse da zero e sono a
d~e
a due distinti).
n
Per provarlo, in R , approssimiamo S mediante un sistema 'i
x
8'
i
1
n
",p(x, ... , x ) ,
i
= 1, 2, .•. ,
i
n
dove i P (x) sono polinomi reali. 8e i ooeffioienti dei P
i
sono gene-
rioi (nel senso della ~eometria algebrioa) i p~nti singolari sono isolati. Nell'intorno di P soriviamo 8'
Faooiamo un' al tra approssimazione, oon Qi ;;; 0. nell' int~rn~ di P . Per l'ipotesi di atabilita .i
so ohe x
= ajx i
j
str~tturale
. La dimostrazione
e
8
e
qualitativamente 10 stes-
ooal oompleta.
2. Le soluzioni periodiohe sono isolate ed elementari. (Cio
vuol dire ohe per ogni soluzione periodioa esiate un intorno tubolare N nel quale non vi sono soluzioni periodiohe oltre quella oonsiderata. Inoltre gli esponenti oaratteristioi della soluzione sono a due a due distinti e, salvo quella banale, hanno tutti modulo
f. 1). Una questione .importante per ora aperta
e
esiste un numero
finito soltanto di soluzioni periodiohe?
3. 8e s 6 una soluzione posltivamente stabile seoondo Poisson (avente oi06 intersezione non vuota oon l'insieme dei propri punti w~limite)
dioa ad s
0
allora s
e
un punta singolare oppure una soluzione perio-
infine esiate un'infinita di soluzioni periodiohe asintotiohe (Congettura di Andronov).
4. 8e 8 ammette una opportuna mis~ra invariante su Mn allota esiste un'infinita di soluzioni periodiahe. 8egue ahe un sistema di Hamilton oon un numero finito di solu-
154
- 5 L.Markus zioni periodiche non pub eaaere strutturalmente stabile.
5. Ogni traiettoria che non sia nomade
e
una traiettoria cen-
trale. Per chiarire questo punta riccrdiamo ohe seoondo la teoria di Birkhoff una traiettoria s di S si dice nomade (wandering) se e81ste un intorno tubolare di s ohe a part ire da un oerto istante non interseca mai piu la propria immagine. L'1nsieme
dei punti ap-
(0 1
n
partenenti a traiettorie nomadi e aperto in M , quindi il oomplementare M1
n
=M
n
e 1 ,oostituito
dai punti non nomadi
e
ohiuso e quin-
di (essendo M oompatta) oompatto. RifetendoLa oostruzione a part ire dal sistema dlnamioo oonsiderato in M1 ' con La topologia relativa, si ottiene una successione decresoente di insiemi oompatti yn:)
M1 :>
M2 => M3 .)
la cui intersezione M , non vuota,oostituisce quello ohe Birkhoff r
ohiama oentro del sistema dinamico. In esso si trovano i punti singolari, Ie soluzioni periodiohe e in generale Ie soluzioni stabili secondo Poisson. La costruzione di Yr
e assai oomplicata in generale.
L'affermazione fatta all'inizio di questa nO e la seguente : Se il sistema S
e
struttura}mente stabile oocorre un solo passo per otte-
nere il oentro poiohe Ml
= Mr
APPLICAZIONI. 1. Consideriamo il flusso lungo Ie geodetiohe d'una superficie ohiusa a curvatura negativa costante. Questione : Un tale sistema dinamico ha stabilita strutturale? 2. Problema di H.Seifert. Consideriamo un oampo di vettori sulla sfera
s3; senza punti
singolari. Esiste sempre una soluzione periodioa? Nel caso della stabilita strutturale Ia risposta e affermativa. Infatti se s1 segue
155
~
6 L.Markus
una traiettoria i1 oa oppure
3.
e
SUg
insieme w-1imite b una traiettoria periodi-
i1 limite d'una suooessione di traiettorie periodiohe.
Genera1i~zaz1one
del teorema di Poinoare-Bendixon.
Dato in un toro solido dt R3 un oampo di vettori senza punti singo1ari tali ohe que1li assooiati a11a superfioie penetrino tutti ne11 1 interno eaiate una traiettoria periodioa? La riapoata b affermativa ne1 oaao one vi sia stabi1its struttura1e.
156
- 7L.Markus
BIB L lOG R A F I A 1. A. Andronov - L. Pontrjagin,
SystS'II68 grossiers, Dokl. Akad. Nauk
SSSR, 14, 247-251 (1937).
2. G. D. BirkhoU,
3.
Dyn"mica~
Systems, New York (1927).
H.De BaggiB, Dynamica~ systems with stab~e structures, Contrib.
to the Theory of nonl. oBoil1., 2, 37-59, Prinoeton (1952). 4. L.MarkuB, GLobaJ structure ,ofordin"ry differenti"~ equ"tions ~n
the
'p~"ne,
Trans. Am. Math. Soo. 76, 127-148 (1957).
5. L. Markus,StructuraL Ly st"b the Am. Mat h . So 0
."
(
~e
differentiaL systems, Notioes of
1959 ) .
6. L.MarkuB,Periodic soLutions and invaria.nt sets of structuraHy
:stabLe differentiaL 'equations,Proe. of the Oongress on Non1. diff. eq:., Mexioo (1959).
7. L.Markus, .Inv"ri"nt :sets :of structuraHystabLe differentiaL sy'st61ls, ·Proe. 'Nat. Aoad.SoLUSA (1959). 8. 'L.Markus,StructuraHy st"He ·differenti"Lsystems, Ann. of Math.
(1961). 9. :1i.·M.Peixoto, On structuraL stability, Ann. of Math., 69, 199-222 (1959}. 10. M.M.'Peixoto,·S'ofa.e ' exaflpL'es
; pn n~di'll8'fl,sionaL
structuraL stabdi-
ty, Proe. Nat. Aoad. Soi. USA, 49, 633-6360959). 11. G. Sansone - R. Conti, Equa.sioni differensiaLi non Liuari, Roma
(1956}. 12.
H~Seifert,
CLosed integraL curves in 3-space and isotropic two-
dimensionaL dejormations, Proe. Am. Math. Soo., 1, 287-302 (1950). 13. S. Smale, StructuraL Ly st"b Le differentia.L systems on cLosed
3-manijoLds, Proe. of Congress on non1. diU. eq., Mexioo (1959 ) . 157
CENTRO INTERNAZIONAI,E MATEMATICO ESTIVO (C.I.M.E. )
G. PRO D I
TEOREMI ERGODICI PERLE EQUAZIONI DELLA IDRODINAMICA
ROMA - Istituto Matematico dell 'Universit8 ;: 1960
159
TEOREMI ERGODICI PER LE EQUAZIONI DELLA IDRODINAMICA di
In questi ultimi anni mente,
[a)
(1)
G. PRODI
e stato dimostrato ([4} e, suocessiva-
ohe 11 problema ,misto (nel senso di Hadamard) per le
equazioni di Navier-Stokes in due variabili spaziali
e univocamente
risolublle "in grande". Questo risultato permette di impost are problemi di comport amento asintotioo per t - too : stabilita, esistenza di soluzioni peri 0diohe, teoremi ergodioi. Si tratta dei problemi che hanno il maggior interesse anohe dal punta di vista fisico. In quello che segue svolgeremo qualohe oonsiderazione su questi problemi; in molti punti non potremo che segnalare emettere in evidenza le diffioolta ohe 8i incontrano. Esaminati i (poohi) dati che l'analisi attualmente ci fornisoe intorno alle equazioni di Navier-Stokes riohlameremo aloune nozioni di carattere generale relative all'esistenza dl misure invarianti. Ceroheremo poi di adattare 901 nostro oaso 190 teoria di Kriloff e Bogoliuboff [3] ed enunoeremo alcuni risultatl ohe si possono oOSl ottenere.
§ 1.- Indiohiamo oon
n un
insieme aperto limitato del piano, oon
u un vettare reale dl oomponentl
u. (j J
------------..--(1)
=
1,2), funzioni del punta
Le rioerohe qUi esposte sono state finan~iate dall'Air Research and Development Command, United States Air Force, can oontratto AF 61 (052) - 414.
161
- 2 -
G.Prodi Indicheremo oon LP(O) 10 spazio dei vettori oon ccmponenti a p-esima potenza sommabile, oon la norma lui dove luCx) I indioa 11 modulo di u(x). Se zio ha struttura hilbertiana (reale),
10
dotto soalare (f, g);:
f fig i
~
t.: (n.)
;: {f lu(x) I P dx}i/p,
0
p ;: 2 , questo spamediante ilpro-
ind'~iduabile
dx . Soriveremo If I in luogo di
If IL~(R;)' Siano poi, f t g
vettori definiti in 0 aventi derivate prime
(in senso generalizzato) a quadrato sommabile .in og _i ox.
;: I r ~
«(f,g)) 8i'l
Soriveremo
2
, II f II ;:
«f,r)'). ox. J J .)f(O) la varieta dei vettori· indefinitamente differenzia- · fl ij
dx
rr.
bili, a divergenza nulla e nulli fuori di un compatto contenuto in
n.
Ni (0) 1a
Sia N(O) la ohiusura di Jf(O) secondo la norma I "
phiusura di .)f(0) seoondo 1a norma
II II.
1
Gli elementi di N (0) so-
no, in un oerto senao, vettori nulli sulla frontiera di 0 . Indichiamo .oon Co 11 .ooefficiente di immersione lui
!lull-i.
Rioordiamo la importante disuguagUanzs ..dd. .Ladyzenskaia (valida per un oampo O. qualsiaei ) .: .1.uI 2,~ .L 1
Se u, v, 11' (; N (0) poniamo
(lli
II 'ull
< .~.,1/2 Iul
. b,()1.~.");:
I o ijr
u
(vedl' [4J).
OVj
i
-0;[
Wj dx .
i
Si ha facilmentelb(u,v,w)1 ~ .Iul 4 ' IIvll 111'1 ~ . • L (n') L ~.n,} p.
Dato uno spazio di Banaoh B , indlaheremoinfine oon L 10 spazio delle tunzlbhi definite nell'intervallo (O,T),
in
B ,
(O,~;B)
oon valori
a p-esima potenza sommabile.
Le equazionl ohe oonsldereremo .con Ie relative oondizioni al oontorno,
~i
esprlmono )
' ln i t~rmini
cla*sioi, in questo modo :
162
- 3 G.Prodi 'Ou ,
-.l
'Ot
( 2)
div u
(3)
u(X)
'Ou ui - j 'Ox i
~
+
i
-
jJ, 142 u j
=-
'Op
-
f
t
'OXj
j
(j = 1,2)
=0 =0
per
x E
TodD) .
Qui la inoognita p ha i1 signifioato di pressione, jJ, indioa il ooeffioiente di visooeit •. leL seguito noisupporremosempre f = (fl' f 2 ) junsione de LLasoLa, x . Questo per metteroi in una 6ituazione di analogia, per quanto • possibile, oon la teoria dei sistemi autonomi . ordinari. 8upporremo
2
f ' L (0); diremo ohe \lna funziorie. u{t},. iooalmente
.limitata oome funzJoneoon valori in N(O), looalmente a quadrato sommabile oome funzione oon valori in N1 (Ol, • soluzione (debole) delle equazioni di Navier-8tokes se vale la relazione (l)
(4)
J
o
{-(u(t),v'(t)) tjJ,«u(t),v(t)))tb(u(t),u(t);v{t))} dt = (l)
=J
o
(f(t},v(t))dt
per ogni funzione vet) .ohe : 1
(a) sia oontinua oome funzione di t .oon· valori in N (0); (b} si annuiLii fuori d1 un .intervallo
7" ....
7'", oon 0 < 7"
< 7'''< too;
2
(0) abbia derivata rispetto a t v'(t) & L (0, till; N(O)).
8i dimostra ohe : 1) (Vedi [9]) le soluzion1 della (4), eyentualmente oorrette su un
insieme di valori di t d1 misura nulla" sono funzioni oontinue di t nelld spazio NCO). 2)
(Vedi . [9]) la funzione lu(t)
I2 e
assolutamente oontinua · e · soddi- ,
sfa all'equazione differenziale
(5)
1
d
2
dt
Iu ( t ) I
2
+ jJ,
2
II u ( t ) 11= 163
(f, u ( t ) )
- 4 -
G.Prodi 3) (Vedi [7], [8]) Assegnato oomunque un valore una ed una sola soluzione ohe 10 assume per t in tutto l'intervallo 0"'"
= Tt
Posto u(t)
u
00
Uo
=0
E N(O), esiate ; easa
e
definita
•
,T
definisoe una applioa~ione oont~nua: t (Ot-oo) X N(fl) ... N(O). (Nonmi e riueoito di vedere se 11.1 oontinuit •. in questione possa
o
sus~isters
in modo uniforms).
Dalla (5) si rioava faoilmsnts
( 6)
~
1 u (t) 1
JJ. -::Tt 1u
o
1
e
On
t
8i ha dunque, per ogni soluzione, 11.1 relazione
i1iii
( 7)
lu(t)
t-rtoo
1
2 JJ. -1 1f1. Cn
~
Anoora dalla (5) si rioava 11.1 dieeguaglianza JJ.J
r
lIu(t)1I
o
Applioando 11.1 (6) si ha 7
J
(8)
o
II
Ilu(t)
2
2
1
dt
112 .
8i ha dunque dalla (10) (1)
8i pub dimostnare che l'insieme 8 delle soluzioni costanti ha ,questa proprieta : per ,ogni ,suo punta u esiste una ,variete. line8,re ad un numeno finito"di dimensioni ohg EI, in un ,oerto sensa, "tan,gente" ad 8 Gin Nl(O) ). in un intorno di u , 8 si ,proietta ortogona1mente (sempre in Nt(O) ) su , questavar~eta senza sovrapposi,zioni. Non e detto perc ,ohe questa proiezione ,sia "su".
165
- 6 G.Prodi
1
d
2
dt
Da questa
e
fao11ededurre ohe, per t ... t a').1 w( t) ... 0
2
too
,
e ohe
Ilw(t)11 dt < ta'). Dunque, se e 21/20 ,u- 1 1Iull < 1 (11 ohe 900n 0 -1 / 2 2 ~2 oade se e I f I < 2 la soluzione u e stabile; segue anohe ,u 0
J
n)
ehe essa
e
unioa.
SUl problema della stabilita ,delle soluz10ni oostantinon si oonosoe ,nullapib di queste se,plioi osservazioni ohe riguardano soltanto il oaso di u "piooolo" (e ;ohe si trovano in blema della stabilita
e
legato strettamente a qUel10
15l).
Il pro-
del1Iunieit~;
vedremo suooessivamente oome questo problema 6 ,legato 901 ,problema ohe oi interessaprinoipalmente.
12.-
Riohiameremo qui alouni risultati ,relativi al1a teonia de11a
,misura sugli spazi ,metrioi e 901190 teoria ergodioa 'olassioa, 90110 soopo prinoipllle di adattarli 901 nostro ,pr.oblema. SiaE uno spllzio ,metrioo sepllrabilecompleto,sia ~(E) ,la famiglia dei boreliani ,di E . Si dioe ,misur.a di pr.obabilitauna fun2lione non negativa definita ,per ,un ,CT-oampo q ... additiva,
tale ohe
E' 1 e ,m(E) = 1
g ,di
ins1em1 di ,E ,
. Noi in1ienderrelllo ,selll'/lr;e"
,ina Hr.e ,che ;m ,sia" i £ 'GollpLetalllento ,a,LLa, ;Lebesgue Idi luna, .misur.a, ,definita, in G!>'(E). Dalla :miBura
111
si ,ottiene ,un integrals, aon ,190 consueta de-
,finizione. Bannemo ,M(r)
=.J
r(u) dm(u) . E
Indiahiall10 aon
e(j\) ,10 ' spuio delle ,run2lioni ,continuee
,1imitate definite in E " .oon , lanorma Irl ~(E) = sup Ir(u) I . v ,u EE ,LI integrale definisae dunque ,in t,(E) un funzionale ,M(r) ,linear e , ,continuo non negativo e ,ahe, inoltre, gode di questa ,propnieta Cdi Daniell) : per ogni suooessi one{r h } di funzioni continue non
166
- 7G.Prodi negative oonvergenti a zero in modo monotono si ha: Vioeversa supponiamo di assegnare in
lim MCf ) = 0 n n_ClJ ~(E) un funzionale ohe gode
di questa proprieta (diremo: un integrale astratto); esso ai puo (1) prolungare oon un prooedimento ,ben nota . ; ai potranno .allora dire miaurebili gli insiemi ,Ie oui ,funzi.oni ,oaratt.enistioherisu1tano sommabili. Se aggiungiamo ,1a ,oondizione di nonmali3zazione : M(1)
=1
Goon 1 indiohiamo. qui ,la funzione ,ohe 'e uguale ad 1 au E), abbiamo una oompleta equiyalenza tramisura (~i probabilit1»
ed integrale.
Partendo da una misura m e definendo un integrale M, attraverso ques.to riotteniamo lamedesima ,misura di parte;nza.
Coa~,
partendo da
un integrale M ,e paasando agli insiemi ,m:Lsura.bili, possiamo riottenere
l'~ntegrale.
La oonsiderazione di un integrale al posta di una misura, e spesso pib comoda. Questo vale ad
ese~pio,per
istituire una topo-
logia nell'insieme della ,misura. Noi dotere,mo l',insieme dellemisune della topologia oheviene subordinata in eeso dalla topologia debole , dello'spazlo
"'.
u . (Identifioando ,lemisune ,oon i relativi' integrali,
ohesono elementi di 211.1
Clf ).
E' ,impo!1tante avene ori.teri di oompattez-
,per ,Ie ,misure. ,La sfera unita!1ia ,dello spazio
~lf(E) ,e, in ,ogni
oaso, oompatta (sempne ,ne:j.la ,topologia debole). -Ora, se ,10 spazio E e oompatto, tutti i t'unf'ionali ,nonnegativi .sono ,misure ~infatti, in virtb del teorema di Dini, sussiste ,la proprieta di Daniell).. Po :Lohe la afar a uni tar ia dell 0 apaz:Lo
~*( E ~ J ,ne 11 a to po log i a de b 0-
le, possiede una baae numenab:He di intorni, si deduoe ,ohe do, ogni
successione
{m }
n
,di!llisur:e di probabi ~ita si ,puo estrarre una
aessione partiaLe aonver,ente v6rso;una,misura di
,suc-
~robabiLita.
Ti)---------Vedere ad esempio, Loomis, Abstraot harmonio analysis, cap.nI.
167
(1953)
- 8 G.Prodi Su questa risultato si fonda 111. dimostrazione dell'esistenza di una misura invariante rispetto ad un gruppo {T } di trasformat zion1 (-00< t < + (0) data d!\ Krylotf e Bogoljuboff [,]. Noi abbiamo bisogno di estendere, in qualche ,modo, 11 risultato enunciato 11.1 CaSO in cui .E non sill. oompatto. Da un risultato di Ulam (richiamato in [11]) s1 ha che, assegnata .in uno spazio E .metrico, completo, separabile, unam1sura m " s1 puo trovare un compatto D tale che ;m(E-D) ~
per ,ogni E •
E
> 0
Noi considere-
remoin E una famiglia di misure di probabilitb che soddisfino a questa condizione
unijo~memente.
Precisamente : sia assegnata u-
na successione {Dn} (n= 1,2,3, .•. ) di insiemi compatt:!:, can D C D . S1&
n
n+1
m la
famiglia di tutte le misure di probabilita
m che soddisfano 11.1111. oondizione : m(E-D ) ~ n
1 n
S1 ha Il.'llora :
La
famig~ia
JY(,
e
compatta, neHa
topo~ogia
Baster1:r. dimoatra:re che lafamiglia It[, so di
e
introdotta.
un aott01nsiemechiu-
'c*(E). Indioando aempre con 'Mil funzionale corrisponden-
te alla .misura ,m"
msi
:la .condizione che1ndividua la famigUa
traduce nella seguente : presoun interon > '01 per ,ogni funzione ¢ , continua, nulla in D
J
n
S1& ora F un elemento di di )tl.
.con 0
~ ¢ ~
1
si ha M(¢) < -
2.. n
'e~(E) appar.tenentealla chiusura
Evidentemente F sara non negativo, tale che F(,l') = 1
Inoltre soddisfere. alla condizione : F(¢) - 0 e preso un dominio D
m
~ntero m tale che ~ < ~ , consideriamo il
. Ciascuna delle funzioni ¢
168
n
m 2 puo essere deoomposta in
- 9 G.Prodi que s t
0
modo : ¢
l'1nsieme D m
n.
¢t > 0 ¢" > 0 CP';: ¢ n n ,n 'h
;: ¢' + ¢" ., dove n n
Imp~rremo
.inoltrela oondizione
S U 1-
;:
Poiche, per il teorema di Dini, ¢
converge unifor-
n
memente a zero Sl.\ Dm esiste un intero no tale ,che, ;per ogni n > no s1 ha
.II¢~ II C(E)
;: II¢n II e(D m) ( :
Per
n > no si /lvrs. allora
Si ha inoltre
NdLa topoLogia indotta in Jrl,da
tlt(E)
ogni punto rpossiede
,una ,base numer;ab i Le di intorni. Inhttfi, s h i t
nk
} (n, k ;: 1,2, ... ) un sistema di funzioni oon-
tinue soddisfacenti a queste condizioni : a) Ilf ·gni ' n ·fissato 'Ie restrizioni di f a D nk '
n
I '(M'
).1
- Ml(.)(fnk .
G(Dn)
II f nk II 1I~( Dn ).
0)
m. ,
II~ 1; b) per
gli insiemi formati dalle misurem 1 ('-r
0 -
(k ;: 1,2, ... ) costi tuiseo-
no un insieme denso nella sfera unitaria di
Se m*"f
nk
;
Em,
tali che
(n,k ,r;: 1 , 2,... ) cos t1' tulscono ' .un SlS . t ema
fondamentale di intorni. Possiamo ooncludere
neLLe ipotesi fatte, da ogni successions {m·n-} dimisure appartenenti aLLa famigLia zia~e
nu
si pu6 estrarre unasuccessione par-
convergente. Supponiamo ora ohe sia assegnata una famiglia {T t } di trasfor-
mazioni dello spazio Ein se. Per pat ere camprendere ilcasoche ci
inter~ssa,
supporremo 'ohe ·es.sa sta ·dei':i:nita 'per '0
~ ·t
(+co
e
che costituisoa un semigruppo. (Tutto ,fa 'pensare ·che ole trasformazioni Tt definite nel § 1 non abbiano inversa continua). Supp o rremo (ofr. § 1, propriet1l. ,3) ) e he Tt definisca una applioa z ione continua (0 r- +(0) X E ~ E . Supponiamo assegnata in E una ,misura
169
- 10 G.Prodi di probabilit. m . Essa vi ene , mutata dalla Tt (per ogni valore t .? 0 fissato) in unamisura ,m, seoondo la relazione
t
If(u)dm t valida per ogni funzione Se una funzione f finita : f' (u) = f(Tu)
m
e
e
= If(Ttu)dm f G 1
(T u ) dt = '1 0
t
Indiohiamo oon m7 la misura definita dall i integraie
=
M (r) 7
1
7
f
f(T u )dt •
t
o
0
Dimostriamo ohe questa famiglia soddisfa alle oondizioni del .oritenio di oompattezza introdotto. nel §. 1, almeno per7 ~ 1 . ·Oonsideriamoinfatti una qualunque ·funzione .oontinua .tf; A}
o~n 0 . Then F (A)
n
n Fm(A)
F (A) , n
~
if
= if;
nt min, m = 0, 1, ...
n = 0, 1,... .
n
This is an immediate
oonsequenoe of the definition of oonditional probability. (iii)
f
We reoall that, by definition,
c
fF
n
P (AI ~) dP = P (A (l C)
(A)
C€ ~
if
PCAI ~ ) dP = P(F (A) () A) < e
n
n
Thus
-A P(F
n
(A))
for eaoh n = 0, 1, ... and f.. > 0 Q) (iv) I assert E(A) .= U Fn (A) f) A . For, if x fA, we have n:=o
r(Q,! en) (x)= - log
p(AI
t: n ) (x)
(- log P (AI
sup o~n
