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G}
{(p : (R -+ G} such that {J(p = G) {&P = G}.
Let G be a topological group. Then: (1) If G is monothetic or solenoidal, then G is abelian. (2) G is monothetic if and only if there exists x E G such that [x° I n E 5]
4.50. REMARK.
is dense in G. 4.51. REMARK.
Let X be a compact uniform space. Then the following
statements are equivalent: (1) There exists a { discrete } { continuous } flow on X under which X is an almost periodic minimal orbit-closure.
TOPOLOGICAL DYNAMICS
40
[4.51]
(2) There exists a group structure of X which makes X a {monothetic} {solenoidal } topological group. 4.52. THEOREM. Let G be a compact abelian group. Then the following statements are pairwise equivalent: (1) G is monothetic. (2) G is separable and G/K is monothetic where K is the identity component of G.
(3) G is separable and if H is an open-closed finite-indexed subgroup of G, then G/H is cyclic. (4) The character group of G is algebraically isomorphic to a subgroup of the circle group. PROOF. Cf. Anzai and Kakutani [2]. 4.53. REMARK.
Let G be a compact connected separable abelian group.
Then G is monothetic. Let G be a compact abelian group. Then the following statements are pairwise equivalent: (1) G is solenoidal. (2) G is separable connected. (3) The character group of G is algebraically isomorphic to a subgroup of the line group R. PROOF. Cf. Anzai and Kakutani [2]. 4.54. THEOREM.
4.55. THEOREM.
Let X be a compact uniform space containing more than
one point and let X be an almost periodic minimal orbit-closure under a continuous flow (X, (1, 7r). Then X is not totally minimal under (X, (R, 7r). PROOF. Let x E X. By 4.47 we may suppose that X is a topological group such that 7rs : 61 -* X is continuous homomorphic. By the theory of characters
there exists a continuous homomorphism p of X onto the (additive) circle group C. There exists t E (R such that the subgroup of C generated by xtcp is finite. Since [x(tn) I n E g](p = [(tn)ir.,p I n E 9] = [(tar=(o)n I n E J] = [(xtcp)n
n E J], the orbit of x under the discrete flow generated by ir` is not dense in X. The proof is completed. 4.56. REMARK. The only universally valid two-termed implications among the almost periodicity properties are the obvious ones. These implications are summarized in Table 2. (See Part Two.)
4.57. REMARK. Let (p be a function on a topological group T to a uniform space Y. Then the following statements are pairwise equivalent: (1) cp is {left} {right} uniformly almost periodic. (2) If a is an index of Y, then there exists a left syndetic subset A of T such
rtv) E a) . (3) If a is an index of Y, then there exists a right syndetic subset A of T'
that r (E T and t E A implies { ( r r p , t r o p ) E a } { (rcp,
such that r E T and t E A implies { ( r c p , t r i p ) E a } { (rrp, rtlp) E a}.
ALMOST PERIODICITY
14.59]
41
(4) If a is an index of Y, then there exists a compact subset K of T such that t E T implies the existence of k E K such that r (E T implies { (krrp, trop) E a) { (rkcp, rtp) E a}. TABLE 2
T almost periodic
T locally almost periodic
) T locally almost periodic
at x
T weakly
almost periodic
T locally weakly almost periodic
T locally weakly almost periodic
at x
I
uniform almost periodicity properties
I
T poi se almost periodic
T almost periodic
pointwise almost periodicity properties
almost periodicity properties
at x
at a point
4.58. THEOREM. Let cc be a continuous {left} {right} uniformly almost periodic function on a topological group T to a uniform space Y. Then cc is {left} {right} uniformly continuous and bounded. PROOF. Suppose (p is continuous and left uniformly almost periodic. We show cp is left uniformly continuous. Let a be an index of Y. Choose a symmetric index ,6 of Y such that $8 C a. There exists a compact subset K of T such that t E T implies the existence of k E K such that (krcp, trop) (E,6 (r E T).
Select a neighborhood U of e such that kUcp C kcpe (k E K). It is enough to show that t E T implies tUcp C trpa. Let t E T. There exists k E K such that (krcp, trip) E ,B (r E T). Hence tUso C kUcp$ C
C tcc$ C tcpa and tUcp C trpa.
We use 4.41 to show that ip is bounded, that is, the range Tip of cp is totally bounded. Let a be an index of Y. There exists a compact subset K of T such that t E T implies the.existence of k E K such that (krcp, trip) C a (r (E T). Then Tip C Kioa. The proof is completed. 4.59. THEOREM. Let (4), T, p) be the {left} {right} uniform functional transformation group over a topological group T to a uniform space Y and let cp E -1). Then: (1) The orbit pT of (p is totally bounded if and only if (p is {left} {right} uniformly almost periodic.
_
(2) If Y is complete, then the orbit-closure cpT of ip is compact if and only if p
is { left } { right } uniformly almost periodic. PROOF. Use 4.43 and 11.05.
42
TOPOLOGICAL DYNAMICS
[4.601
4.60. REMARK. Let 'p be a complex-valued function on a discrete group. Then (p is { left almost periodic } { right almost periodic } { almost periodic } in
the sense of von Neumann if and only if rp is { left } { right } { bilaterally } uniformly
almost periodic. (Cf. von Neumann [1].) 4.61. THEOREM. Let 'p be a continuous function on a topological group T to a uniform space Y. Then the following statements are pairwise equivalent: (1) rp is bilaterally uniformly almost periodic. (2) cp is bilaterally uniformly almost periodic with respect to the discrete topology of T.
(3-4) p is { left } { right } uniformly almost periodic and { right } { left } uniformly continuous.
(5) If a is an index of Y, then there exists a finite partition 8 of T such that E E 8 and t, s E T implies tEs'p X tEs'p C a.
PROOF. We show (1) implies (2). Let 'p be bilaterally uniformly almost periodic and let (4>, T, p) be the {left} {right} uniform functional transformation group over T to Y. By 4.58, (p E 4), and by 3.43, 'p is an almost periodic point under (4), T, p). It follows from 4.43 that T is discretely almost periodic on rpT, whence, by 3.43, (p is { left If right) uniformly almost periodic with respect to the discrete topology of T. Clearly (2) implies (1). By 4.58, (1) implies (3) and (4). We show 1(3)) { (4) } implies (5). Assume 1(3)1 {(4)J. Let ('i, T, p) be the (left) { right I uniform functional transformation group over T to Y. By 4.58,
(p E T); by 3.43, cp is an almost periodic point under (4,, T, p); and by 4.43, (PT is totally bounded. Let a be an index of Y and let a. (cf. 11.01) be the correspond-
ing index of 4). By 4.38, there exists a finite partition 8 of T such that E E 8 and ¢ E rpT implies VE X ¢E C a*d . Let E E 8, let t, s E T and let Then PE X ,/'E C at implies tEscp X tEscp C a. This proves (5). We show (5) implies (2). Assume (5). Let a be an index of Y and let 8 be the corresponding finite partition of T. Let K be a finite subset of T such that E E 8
implies K n E # 0. Let t E T. Then there exists E E 8 and k E K such that t, k E E. Let r E T. It follows that ETlp X Ercp C a, whence (IcTcp, trop) E a, and rEEp X rE(p C a, whence (rkcp, rt(p) E a. By 4.57, 'p is bilaterally uniformly almost periodic with respect to the discrete topology of T. The proof is completed. 4.62. REMARK. Let cp be a continuous function on a topological group T to a uniform space Y and let the left and right uniformities of T coincide. Then 'p is left uniformly almost periodic if and only if 'p is right uniformly almost periodic. In such an event the words "left" and "right" may be omitted.
Let T be a topological group, let Y be a uniform space, and be a finite class of continuous bilaterally uniformly almost periodic functions on T to Y. Then the function class is bilaterally uniformly almost periodic. 4.63. THEOREM.
let
ALMOST PERIODICITY
[4.66]
43
PROOF. Let ('F, T, p) be the {left} {right} uniform functional transformation group over T to Y. By 4.61, 4) C'F. Hence, (PT = U, e 4, (pT is totally bounded. By 4.38, ('F, T, p) is almost periodic on 4T and therefore, by 3.45, the function class (PT, which contains -1), is {left} {right} uniformly almost periodic. The proof is completed.
4.64. THEOREM. Let T be a topological group, let Y be a complete uniform space, let n be a positive integer, let 0 be a continuous n-ary operation in Y, and let (p, , , (p,, be continuous bilaterally uniformly almost periodic functions on T to Y. Then 0((p, , , pn) is a continuous bilaterally uniformly almost periodic function on T to Y. PROOF.
Let a be an index of Y. Now E = TV, X ... X Tcn is compact
by 4.60 and therefore 0 is uniformly continuous on E. Hence, there exists an , n) with (x; , y;) E a (i = 1, index /3 of Y such that x; , y; E T(p; (i = 1, , xn), 0(y, , , y.)) E a. By 4.63, there exists a 2, , n) implies (O(x,, left syndetic subset A of T such that r E T and t E A implies { (T(p{ , tr(p;) E ,9 , n) 1. Hence, T/ E T and t E A , n) } { (r(p; , tr(p;) E 0 (i = 1, (i = 1, implies 0(rt(p, ,
{ (0 (7-(P., ,
...
,
O(tT4,l
,
...
tT con))
E a) 1 (0(T(pl
,
...
, T(Pn),
, rtp )) E a}. The proof is completed.
4.65. THEOREM.
Let (X, T, 7r) be a transformation group such that X is a
compact uniform space and T is almost periodic, let Y be a uniform space, let p be a continuous function on X to Y, and let a E X. Then the function 7ra(p : T -* Y is bilaterally uniformly almost periodic. PROOF. We show 7ra(p is right uniformly almost periodic. Let a be an index of Y. Choose an index /3 of X so that (x, y) E ,B implies (x(p, y(p) E a. There exists a left syndetic subset A of T such that xA C x,8 for all x E X. If r E T and t E A, then (r7ra , Tt7ra) = (aT, art) E / and (T7ra(p, Tt7ra(p) E a. By 4.39, Ira is left uniformly continuous. Hence, 7ra(p is left uniformly continuous. By 4.61, 7ra(p is also left uniformly almost periodic. The proof is completed. 4.66. LEMMA.
Let X be a compact uniform space, let a be an index of X,
and let Y be a uniform space which contains an arc. Then there exists an index ,l3 of Y and a finite set 4' of continuous functions on X to Y such that x, , x2 E X with (x,(p, x2(p) E ,e ((p E 4') implies (x, , x2) E a. PROOF. Let E be an are in Y, let yo , y, be the endpoints of E, and let Q be an index of Y such that (yo , y,) /3. Choose a symmetric open index y of X such that y3 C a. Select a finite subset A of X such that X = U.GA ay. Since X is normal, for each a E A there exists a continuous function 4a on X to E such that x(pa = yo (x (E ay) and x(pa = y, (x (E X - ay2). Define 4) = [(pa a E A]. Let x, , x2 E X with (x,(p , x2(p) E /3 ((p E 4'). Assume (x, , x2) a. aye since otherwise x2 E aye, Now x, E ay for some a E A. It follows that x2
TOPOLOGICAL DYNAMICS
44
[4.661
a E x,' , x2 E x,y3 and (x, , x2) E y3 C a. Hence x,cpa = yo , x2tpa = y, and (yo ,
y,) = (x,cpa , x2c .,) E ,6. This is a contradiction. The proof is completed.
4.67. THEOREM.
Let (X, T, 7r) be a transformation group such that X is a
compact uniform space which is minimal under T and let Y be a uniform space which contains an arc. Then the following statements are pairwise equivalent: (1) T is almost periodic. (2) If rp is a continuous function on X to Y and if a E X, then the function 7ra(p : T - Y is bilaterally uniformly almost periodic. (3) If p is a continuous function on X to Y, then there exists a E X such that the function 7rarp : T -* Y is right uniformly almost periodic.
PROOF. By 4.65, (1) implies (2). Clearly, (2) implies (3). Assume (3). We prove (1). We first show (I): If p is a continuous function on X to Y and if x E X, then 7rycp is a right uniformly almost periodic function. Let (p be a continuous function on X to Y. By hypothesis, there exists a E X such that 7rarp is a right uniformly almost periodic function. Let a be a closed
index of Y. There exists a left syndetic subset A of T such that (anp, artcp) = (riralp, Tt7r,jp) E a (r E T, t E A). By 3.32, (x(p, xtcp) E a (x E X, t E A). Hence (7-7r. (P, Tt7rep) = (x1- , xTtpo) E a (x (E X, T E T, t E A). This proves (I). Let a be a closed index of X. By 4.66, there exists an index 3 of Y and a finite set 4> of continuous functions on X to Y such that x, , x2 E X with (x,op, x,) E # (,p E I) implies (x, , x2) E a. Choose a E X. By 4.39 and (I), each Trap ('P (E 4))
is a left uniformly continuous right uniformly almost periodic function. By 4.63, the function class { 7ra5e I 'p E -' } is right uniformly almost periodic. Hence,
there exists a left syndetic subset A of T such that (a7', aTtrp) = (T7ra'p, rt7r.-P) E
$ (r E T, t E A, (p E (b). It follows that (ar, art) E a (T (E T, t E A) and, by
3.32, (x,xt)Ea(xEX,tET).
This proves (1). The proof of the theorem is completed. 4.68. DEFINITION.
Let T be a group, let Y be a uniform space, and let
'p : T -* Y. The map p is said to be stable provided that if a is an index of Y, then there exists an index ,6 of Y such that t, s E T with (tip, s(p) E i3 implies (tr', ST(p) E a (r (E T) and (Ttp, rs'p) E a (r E T). 4.69. THEOREM. Let T be a group, let Y be a topological group provided with its bilateral uniformity, and let v be a homomorphism of T into Y. Then 'p is stable. PROOF. if trp (s'p)
Let U be a neighborhood of the identity of Y. If t, s, r E T and
-' E U and (t'p) ' sp E U, then trip (sr'p) -' E U and (Tt(p) -' Tscp E U. The proof is completed. 4.70. REMARK. Let X be a set, let o be a group structure of X, let `U. be a uniformity of X, let 3 be the topology of X induced by 'U., and for each a E `U. suppose there exists,6 E `U. such that (x, y) E $ implies (x o z, y o z) E a (z E X).
ALMOST PERIODICITY
[4.74]
45
Then (X, o, 5) is a topological group such that the bilateral uniformity of (X, o, 3) coincides with `U.. 4.71. THEOREM.
Let T be a group, let Y be a separated uniform space, and
let (p be a map of T onto Y. Then (p is stable if and only if there exists a (necessarily unique) group structure o of Y such that (Y, o) is a topological group, (p : T -* (Y, o) is a homomorphism, and the bilateral uniformity of (Y, o) coincides with the given uniformity of Y. PROOF. The necessity follows from 4.69. We prove the sufficiency. Suppose (p is stable. Define G = ev(p '. We observe that if t, s E T, then the following statements are pairwise equivalent: t(p = s(p; tT(p = sT(p (r E T); Tt(p = Ts(p (T E T); is-' E G. Now G is a group since e E G and from t, s E G it follows that t(p = e(p = s(p and is-' E G. Also G is invariant in T since from t E G and T E T it follows that t(p = e(p, rt(p = Te(p = r(p, Ttr 1 = TT-'(p = e(p, and rtT-' E G. Since (p is
constant on each translate of G, there exists a map 0 : T/G - Y such that ,rip = (p where 7r is the projection of T onto T/G. Now 0 is a one-to-one map of T/G onto Y. Let o be the unique group structure of Y such that 0 is an isomorphism of T/G onto (Y, o). Clearly, (p = 7r( is a homomorphism of P onto (Y, o). To finish the proof we use 4.70. Let a be an index of Y. There exists an index l3 of Y such that t, s E T with (t(p, s(p) E /3 implies (t(p o T(p, s(p o r(p) = (trop, srop) E a (T E T) and (r(p o top, T(p o sop) = (Tt(p, Ts(p) E a (r E T). The proof is completed. 4.72. THEOREM. Let T be a discrete group, let Y be a topological group provided with its bilateral uniformity, and let (p be a homomorphism of T onto Y. Then .p is a uniformly almost periodic function if and only if Y is totally bounded. PROOF.
Use 4.57 and 4.69.
4.73. THEOREM. Let T be a discrete group, let Y be a totally bounded separated uniform space, and let (p be a stable map of T into Y. Then (p is a uniformly almost periodic function. PROOF.
Use 4.71 and 4.72.
4.74. STANDING NOTATION.
For the remainder of this section we adopt the
following notation.
Let N be the set of all positive integers. Let X be a set. If n E N, then X° or X" denotes the nth cartesian power of X. The total power of X, denoted X*, is UnEN Xn. The phrase "finite family" shall mean "nonvacuous finite family". We consider any finite ordered family in X to be an element of X*. We define
the binary operation composition in X* if a = (x1 , b = (y, , , y,n) E X*, then ab = (x1 , , xn , y, ,
,
xn) E X* and if
yn). Composition is associative but not commutative. If n E N and if x E X, then xn or x`n' denotes the n-tuple (x, , x). If (a, c E I) is a finite ordered family in X*, then I
,
46
TOPOLOGICAL DYNAMICS
[4.74]
J,EI a, denotes the continued composition a a,, where I = [c, , , L"] and c, < . < t" . If a C X X X, then a* denotes the set of all couples (a, b) such that for some n E N we have a = (x, , , x") E X" , b = (y, ,
.,y")EX"and
4.75. DEFINITION.
1,...,n).
Let X be a uniform space. An averaging process in X
is defined to be a function µ : X* -* X such that:
(1) If x E X and if n E N, then x"µ = X. (2) If a E X* and if b is a permutation of a, then aµ = bµ. (3) If n E N and if (a, c E I) is a finite ordered family in X", then (a,µ I
c E I)µ = (1 1,., ajµ. (4) If a is an index of X, then there exists an index ,e of X such that (a, b) E (3*
implies (aµ, bµ) E a. 4.76. STANDING NOTATION. Let X be a uniform space, let µ be an averaging process in X, let T be a discrete group and let c : T --> X.
4.77. REMARK. By 4.75(2), if n E N and if (x, E I) is a finite family in X, then (x, c E I)µ is uniquely defined. Similarly, if E is a finite subset of X, then Eµ is uniquely defined. I
Let a and ,3 be indices of X such that (a, b) E (3* implies (aµ, bµ) E a. It follows from 4.75(1) that if x E X and if (x, c E I) is a finite family in x3, then (x, c E I),u E xa. I
I
4.78. DEFINITION. Let a be an index of X. An a-mean of c is a point x of X such that taspµ E xa (t, s E T) for some a E T*. The set of all a-means of
,p is denoted by acp. 4.79. REMARK. If a and (3 are indices of X such that (3 C a, then & C a crd X + crd UaEn (X), - X) > crd UµExvn X which contradicts the hypothesis. The proof is completed. 4.85. LEMMA.
Let X be a set and let 8,
be partitions of X with the same
finite cardinal. Then the following statements are pairwise equivalent:
(1) a C 8, B C F and U a C U B implies crd a 5 crd B. (2) a C 8, B C iF and U B C U a implies crd B < crd a. (3) There exists a common choice set of a and B. PROOF.
Use 4.84.
4.86. THEOREM. Let X be complete and let p be uniformly almost periodic. Then there exists a mean of gyp. 0. PROOF. Let a be an index of X. It is enough by 4.83 to show that acp Let (3 be an index of X such that (32 C a. Choose an index y of X such that
48
TOPOLOGICAL DYNAMICS
[4.861
(c, d) E y* implies (cµ, dµ) E (3. By 4.61(5) there exists a finite partition 8 of T with least cardinal such that tEsso X tEscp C y (t, s E T; E E 8). We observe that if p = (p, c E I) and q = (q, c (=- I) are choice functions of 8, then (tp&rµ, tgscpA) E /3 (t, s E T); for we may suppose that for each c E I the points p, and q, belong to a common member of 8, whence t, s E T implies (tp,&p, tq,scp) C -,y (, E I), (tpsp, tgsp) C y* and (tpsrpµ, tgssoµ) E /3. Let a = (T, I , (E I) be a choice function of 8 and define x = We show x E acp. Let t, s E T. By 4.85 there exists a common choice function b = (Q, E I) of 8 and t-18s 1. I
I
I
Since a and tbs are choice functions of 8, we have (acpi, tbscpg) E /3. Since b and a are choice functions of 8, we have (tbssoµ, tas-pA) E /3. Thus (a(pµ, tas(pjA) E 02
E a and tasrpµ E xa. The proof is completed. 4.87. NOTES AND REFERENCES.
A number of the results of the early part of this section can be found in Gottschalk [3, 6, 7]. (4.48) For related results, see Stepanoff and Tychonoff [1]. (4.52 and 4.53) Cf. Halmos and Samelson [1]. (4.55) This theorem is due to E. E. Floyd (Personal communication).
(4.59) The connection between compactness and almost periodicity of functions was first observed by Bochner (cf. Bochner [1]).
(4.61(5)) This characterization of an almost periodic function is due to W. Maak (cf. Maak [1, 2]) (4.65 and 4.67) Forms of these theorems were originally proved by J. D. Baum (cf. Baum [1]). (4.73) The connections between stability and almost periodicity of functions have been observed and studied many times (cf. Franklin [1], Markoff [2], Bohr [1, C32], Hartman and Wintner [1]). (4.85) Cf. Halmos and Vaughan [3].
5. REGULAR ALMOST PERIODICITY 5.01. STANDING NOTATION.
Throughout this section (X, T, 7r) denotes a
transformation group.
5.02. REMARK. Let x C X. Then: (1) If T is regularly almost periodic at x, then T is isochronous at x. (2) If T is isochronous at x, then T is almost periodic at x. 5.03. THEOREM.
Let x (E X and let T be {regularly almost periodic) { iso-
chronous } at x. Then T is [regularly almost periodic l { isochronous } at every point of xT. PROOF.
Use 3.21.
Let T be a discrete group. Then: (1) If G is a syndetic subgroup of T, then there exists a syndetic invariant sub-
5.04. LEMMA.
group H of T such that H C G. (2) If G and H are syndetic subgroups of T, then G n H is a syndetic subgroup of T, of G, of H. PROOF.
(1) For t E T let gyp, : T/G -p T/G be the permutation of T/G defined
by Ep, = t-'E (E E T/G). Let P be the permutation group of T/G. Define the homomorphism cp : T --> P by tcp = (p, (t E T). Define H to be the kernel of cp. The conclusion follows. (2) If G and H are subgroups of T, then t(G n H) = tG (1 tH (t (E T) whence
T/G (1 H C T/G (1 T/H. The conclusion follows. 5.05. REMARK. Let T be discrete and let x E X. Then the following statements are equivalent: (1) T is regularly almost periodic at x. (2) If U is a neighborhood of x, then there exists a syndetic subgroup A of
T such that xA C U. 5.06. REMARK. Let T be discrete and let x E X. Then the following statements are equivalent: (1) T is isochronous at x.
(2-3) If U is a neighborhood of x, then there exist a syndetic subgroup A of T and t (E T such that {xtA C U} {xAt C U}. 5.07. INHERITANCE THEOREM.
Let T be discrete and let S be a syndetic sub-
group of T. Then:
(1) If x E X, then S is
{ regularly almost periodic isochronous } at x if and
only if T is { regularly almost periodic } { isochronous } at x. 49
TOPOLOGICAL DYNAMICS
50
[5.071
(2) S is pointwise { regularly almost periodic } { isochronous I if and only if T is pointwise { regularly almost periodic } { isochronous 1.
PROOF. From 5.04 the first reading is obvious.
Let x E X. To prove the second reading it is enough to show that if T is isochronous at x, then S is isochronous at x. By 5.04(1) we assume without loss that S is also invariant in T. We first show that if T is isochronous at x, then S. is isochronous at x. Suppose T is isochronous at x. Let U be a neighborhood of x. By 2.10(2), x (E x(T - Si).
Hence we may assume that U n x(T - Sam) = 0. There exist t E T and a syndetic subgroup A of T such that xtA C U. Therefore xtA () x(T - Sam) = 0,
to C SS , t E S. and A C t-'Sx = Sz by 2.10(1). Thus t E S. and A is a syndetic subgroup of S. such that xtA C U. This shows that S. is isochronous at x. We next show that if S is isochronous at x, then S is isochronous at x. Suppose S. is isochronous at x. Let U be an open neighborhood of x. By 2.10(3) there
exists a finite subset M of T such that xM C U and S. C SM'. Let V be a neighborhood of x for which VM C U. There exist t E S. and a syndetic subgroup A of S. such that A is invariant in T and xtA C V. Choose s E S and
m E M such that t = sm'. Define B = S (1 A. Then xsB = xsm-'Bm C xtAM C VM C U. Thus s E S and B is a syndetic subgroup of S such that xsB C U. This shows that S is isochronous at x. The proof is completed. 5.08. THEOREM.
Let X be regular, let T be discrete and let T be pointwise
regularly almost periodic. Then every orbit-closure under T is zero-dimensional. PROOF. Let x E X. We assume without loss that X = xT. Let U be a neighborhood of x. There exists a subgroup A of T and a finite subset E of T such that T = AE and xA C U. By 5.07(2), A is pointwise regularly almost periodic. By 4.08, the class e of all orbit-closures under A is a partition of X. Since X = xT = xAE, e is finite. Hence xA is an open-closed neighborhood of x. The proof is completed.
5.09. THEOREM. Let X be a compact metrizable minimal orbit-closure under T, let T be discrete, let R be the set of all points of X at which T is regularly almost periodic and let R -- 0. Then R is a T-invariant residual Ga subset of X.
Clearly, R is T-invariant. Let `U be a countable base of the uniformity of X such that every element of ti is a closed index and let B be the class of all PROOF.
syndetic invariant subgroups of T. We use the notation of 3.30. If B E B, then by 2.25 the class of all orbit-closures under B is a finite partition of X. It follows that if (3 E V and if B E B, then E(B, 3) is a union of orbit-closures under B and hence E(B, a) is open in X. Since R= no... UBEB E(B, 0) and `U is countable, we conclude that R is a residual Ga subset of X inasmuch as R is represented as a countable intersection of everywhere dense open sets. 5.10. STANDING NOTATION.
a uniform space.
For the remainder of this section X denotes
REGULAR ALMOST PERIODICITY
15.17]
51
5.11. REMARK. The following statements are valid: (1) T is regularly almost periodic if and only if T is isochronous. (2) If T is isochronous, then T is almost periodic.
Let Y be a T-invariant subset of X and let T be regularly
5.12. THEOREM.
almost periodic on Y. Then T is regularly almost periodic on Y. PROOF.
Use 3.33.
Let T be discrete. Then the following statements are equi-
5.13. REMARK.
valent: (1) T is regularly almost periodic.
(2) If a is an index of X, then there exists a syndetic subgroup A of T such
that x E X implies xA C xa. 5.14. INHERITANCE THEOREM. Let T be discrete and let S be a syndetic subgroup of T. Then S is regularly almost periodic if and only if T is regularly almost
periodic. PROOF.
Use 5.04.
5.15. THEOREM. Let X be locally compact, let x E X, let T be isochronous at x and let T be equicontinuous. Then T is regularly almost periodic on xT.
PROOF. By 4.09 we may suppose that X is compact. By 5.12 it is enough to show that T is regularly almost periodic on xT. Let a be an index of X. Choose a symmetric index a of X such that (32 C a. There exists an index y of X such
that y E X and t E T implies yyt C yti3. There exist a syndetic invariant subgroup A of T and s E T such that xsA C xy. If t E T, then xtA = xsAs 't C xys' t-' C xs' t '3, xtA X xtA C 162 and xtA C xt32 C xta. Thus t E T implies xtA C xta. The proof is completed. 5.16. LEMMA.
Let X be compact and suppose that if x, y E X with x 9 y,
then there exist a neighborhood U of x, a neighborhood V of y and an index a of
X such that t E T implies (Ut X Vt) (l a = 0. Then T is uniformly equicontinuous. PROOF.
tEB].
For A C X X X and B C T, define AB = [(xt, yt) (x, y) E A, I
Let # be an open index of X. For each z E X X X - ,Q there exists a neighborhood W, of z and an index a, of X such that W,T (1 a, = 0, whence a,T C
X X X - W. . Choose a finite subset E of X X X such that X X X - 8 C UZEE W, , whencen=EE (X X X - W,) C fl. Define a= naEE a, . Now a is an index of X. Since aT C fl xEE a,T C n,eE (X X X - W,) C 0, it follows that aT C ,6. The proof is completed. Let X be compact. Consider the following statements: (I) T is pointwise isochronous and equicontinuous.
5.17. THEOREM.
52
TOPOLOGICAL DYNAMICS
[5.17]
(II) T is pointwise regularly almost periodic and S is weakly almost periodic for every syndetic invariant subgroup S of T.
(III) T is regularly almost periodic. Then:
(1) I is equivalent to II; III implies I and II. (2) If T is discrete, then I, II, III are pairwise equivalent. PROOF. By 5.15 and 4.38, I implies II. Assume II. We prove I. Clearly, T is pointwise isochronous. It remains to prove that T is equicontinuous. Let x, y E X with x 0 y. By 5.16, it is enough to show that there exist a neighborhood U of x, a neighborhood V of y and an index a of X such that (U X V) T (1 a = 0. Since T is regularly almost periodic at x, there exists a syndetic invariant subgroup S of T such that y (t xS. By 4.24 the class a of all orbit-closures under S is a star-closed decomposition of X.
By 2.30 and 2.37, a is star-indexed. Since y ($ xS, we have xS n yS = 0, that is, xa (1 ya = 0. Choose an index a of X such that xal3 n ya(3 = 0. Let y be an index of X such that y2 C ,8. Since a is star-indexed, there exists an index
5 of X such that xoa C xay and y6a C yay Provide a with its partition uniformity, which induces its partition topology by 2.36. Clearly, a is compact. Let (a, T, p) be the partition transformation group of a induced by (X, T, 7r). Since AS = A (A E a), it follows that (a, T, p) is periodic and thus almost periodic. By 4.35, (a, T, p) is discretely almost periodic and hence, by 4.38, ((t, T, p) is uniformly equicontinuous. If µ is an index of X, then µ* = [(A, B) I A,
B E a, A C Bµ, B C Aµ] is an index of a. Since (a, T, p) is uniformly equicontinuous, there exists an index 0 of a such that (A, B) E 0 and t E T implies (At, Bt) E y*. Since the projection of X onto A is uniformly continuous, there exists an index a of X such that (p, q) E a implies (pA, qA) E 0. Define U = x5 and V = yS. Let x, E U, y1 E V and t E T. We must show (x,t, y1t) a.
Assume (x,t, y1t) E a. Then (x1at, ylat) = (x1ta, y1ta) E 0, whence (x1a, y1(i) E y*, x1a C y,ay and xlay (1 y,ay 0 0. Now xlay C x8ay C xay2 C xai8
and y,ay C yoay C yay2 C ya,6. Since xa(3 (1 ya,3 = 0, it follows that xlay n y,ay = 0. This is a contradiction. This proves I. Now assume T discrete. By 4.37, III implies I. Assume II. We prove III. Let a be an index of X. Choose a symmetric index # of X such that #4 C a. For each x E X there exists a syndetic invariant subgroup A. of T such that xA. C x/3. By 4.25, for each x E X there exists a neighborhood U. of x such that U.A. C xA,z,3. Select a finite subset E of X for which
X = UxEE U.. Define A= n.EE A.. Then A is an invariant subgroup of T and by 5.04, A is syndetic in T. We show that x0 E X implies x0A C xoa. Let x0 E X. Choose x E E for which xo E UU . Since xo E x0A. C xA.$ C x/32 it follows that x E x002 and xoA C xoA. C x,82 C x0,64 C xoa . Hence x0A C xoa. The proof is completed. 5.18. THEOREM. Let X be a compact minimal orbit-closure under T. Then T is regularly almost periodic if and only if T is pointwise regularly almost periodic.
REGULAR ALMOST PERIODICITY
[5.23]
PROOF.
53
Use 4.29, 5.15 and 5.17
5.19. THEOREM.
Let X be locally compact separated and let T be pointwise
regularly almost periodic. Then T is regularly almost periodic on each orbit-closure under T. PROOF.
Use 5.18 and 4.09
5.20. REMARK. The following statements are equivalent: (1) T is weakly isochronous.
(2) If a is an index of X, then there exist a syndetic invariant subgroup A of T and a finite subset K of T such that x E X implies the existence of k E K such that xkA C xa. 5.21. REMARK. Let T be discrete. Then the following statements are equivalent: (1) T is weakly isochronous. (2-3) If a is an index of X, then there, exist a syndetic subgroup A of T and a finite subset K of T such that x E X implies the existence of k E K such that
{ xkA C xa l { xAk C xa } . 5.22. THEOREM. Let Y be a T-invariant subset of X and let T be weakly isochronous on Y. Then T is weakly isochronous on Y. PROOF.
Use 3.33.
5.23. THEOREM. Let X be a compact minimal orbit-closure under T. Consider
the following statements:
(I) T is regularly almost periodic at some point of X. (II) T is isochronous at some point of X.
(III) If a is an index of X, then there exist x° E X and a syndetic invariant subgroup A of T such that xA C xa. (IV) T is weakly isochronous. Then:
(1) I implies II; II, III, IV are pairwise equivalent. (2) If X is metrizable, then I, II, III, IV are pairwise equivalent. PROOF.
(1) Clearly, I implies II; II implies III; IV implies II. Assume III.
We prove IV. Let a be an index of X. Choose a symmetric index l3 of X for which
R3 C a. By 4.21 and 4.24 there exists a finite subset F of T such that x E X implies xT C x,BF-1. Select an index y of X such that x E X and s E F implies xys C xs,B. There exists an index S of X such that S2 C 'Y. Choose xo E X and a syndetic invariant subgroup A of T such that x0A C x0&. By 4.25 and 4.29 there exists a neighborhood U of xo such that UA C x0A8. Select a finite subset
E of T for which X = UE-1. Define K = EF. We show x E X implies the existence of k E K such that xkA C xa. Let x E X. Choose t E E such that xt E U and then choose s E F such that xts E x13. Since xtAs C UAs C x0A6s C
TOPOLOGICAL DYNAMICS
54
[5.23]
x0626 C xoys C x0s(3 and xtAs n x,6 54 0, it follows that xtAs C x,33 C xa. Define
k = ts. Then k E K and xkA C xa. The proof of (1) is completed. (2) Assume III. We prove I. We first show: (L) If U is a nonvacuous open subset of X, then there exist x E U and a syndetic invariant subgroup A of T such that xA C U. Let U be a nonvacuous open subset of X. Choose xo E U and an index a of X such that xoa2 C U. There exists a finite subset E of T such that X = xoaE, -'.
Select an index i3 of X such that (x, , x2) E $ and t E E implies (x,t, x2t) E a. There exists y E X and a syndetic invariant subgroup A of T such that yA X yA C ,9. Choose t E E for which yt E xoa. Define x = yt. Since xA X xA =
yAt X yAt C a and xA n xoa
0,
it
follows
that xA C xoa2 and
xA C xoa2 C U. This proves (L). Let [a,. I n = 1, 2, ) be a countable base of the uniformity of X. Define
V, = X. For n = 1, 2,
, we proceed inductively as follows:
There exists a nonvacuous open subset U. of V. such that U. X U. C a,. By (L) there exist x,. E U. and a syndetic invariant subgroup A. of T such that of x such that x A,. C U . By 2.32 there exists an open neighborhood C U.. V,.+, C U. and It is clear that T is regularly almost periodic at every point of fn=, U
Since f ,-, U.
0, the proof is completed.
5.24. THEOREM. Let X be a compact metrizable minimal orbit-closure under T and let T be isochronous at some point of X. Then T is locally almost periodic. PROOF.
Use 4.30 and 5.23.
5.25. THEOREM.
Let X be separated, let x E X, let there exist a compact
neighborhood of x and let T be isochronous at x. Then T is weakly isochronous on xT. PROOF.
Use 5.02, 4.09, 4.07 and 5.23.
5.26. LEMMA. Let G be a zero-dimensional locally compact topological group such that the left and right uniformities of G coincide. Then every neighborhood of the identity of G contains an open-closed invariant subgroup of G. PROOF. Let U be an open-compact symmetric neighborhood of e. Choose an open-closed symmetric neighborhood V of e such that UV C U. Let H be
the subgroup of G generated by V. Define K= n.,,,, xHx '. Then K is an open-closed invariant subgroup of G such that K C U. 5.27. LEMMA. Let X be a compact uniform space, and let 4 be a zero-dimensional compact topological homeomorphism group of X. Then -D is regularly almost periodic. PROOF.
Use 5.26.
5.28. LEMMA. Let G be a discrete group, let H be a topological group with identity e and let cp be a homomorphism of G into H such that Gyp = H and for
each neighborhood U of e there exists a syndetic subgroup A of G such that Arp C U. Then H is zero-dimensional.
REGULAR. ALMOST PERIODICITY
15.331
55
PROOF. Let U be a closed neighborhood of e. There exists a syndetic subgroup A of G such that A(P C U. Define K = App. Now K is a subgroup of finite index in H. Thus K is an open-closed neighborhood of e such that K C U. The proof is completed.
5.29. THEOREM. Let X be compact, let T be discrete, let 4) be the total homeomorphism group of X and let 4) be provided with its space-index topology. Then
the following statements are equivalent:
(1) (X, T, a) is regularly almost periodic. (2) The closure in 4' of the transition group [7r` : X -' X I t E T] is a zerodimensional compact topological group. PROOF.
Use 4.44, 5.27, 5.28.
5.30. REMARK.
Let p be a continuous homomorphism of a topological
group T into a zero-dimensional compact topological group S and let (S, T, p) be the {left} {right} transformation group of S induced by T under (p. Then: (1) (S, T, p) is regularly almost periodic. (2) S is a regularly almost periodic minimal orbit-closure under (S, T, p) if and only if Tcp = S. 5.31. THEOREM.
Let x E X and let T be discrete abelian. Then the following
statements are pairwise equivalent.
(1) X is a regularly almost periodic compact minimal orbit-closure under (X, T, jr). (2) X is an almost periodic zero-dimensional compact minimal orbit-closure under (X, T, 7r). (3) There exists a unique group structure of X which makes X a topological group such that (X, Try) is a zero-dimensional compactification of T. (4-5) There exist a zero-dimensional compact topological group S and a homomorphism (P : T -* S such that Tip = S and the {left} {right} transformation group of S induced by T under (p is isomorphic to (X, T, a). PROOF.
Use 4.47 and 5.08.
5.32. LEMMA. Let G be a compact metrizable abelian group and let H be the set of all x E G such that the closure of [x" I n E 9] is zero-dimensional. Then H is a dense subgroup of G.
This result is from the theory of Lie groups. 5.33. THEOREM. Let X be a compact metric space with metric p, let cp be an almost periodic homeomorphism of X onto X and let e be a positive number. Then there exists a regularly almost periodic homeomorphism P of X onto X such that x E X implies p(xrp, x¢) < e; indeed, i' 1may be chosen to be uniform limit of a
sequence of (positive) {negative} powers of gyp. PROOF.
Use 5.32.
56
TOPOLOGICAL DYNAMICS
[5.34]
5.34. THEOREM. Let cp be a regularly almost periodic homeomorphism of a two-dimensional manifold X onto X. Then p is periodic.
PROOF. Let a be the partition space of all orbit-closures. By 5.08 the projection P : X -+ A is light interior. It is known (Whyburn [1], p. 191) that a light interior mapping on a manifold is locally finite-to-one. Hence (P is pointwise periodic. It is known (Montgomery [2]) that a pointwise periodic homeomorphism
on a manifold is periodic. The proof is completed.
5.35. REMARK. The only universally valid implications among the regular
almost periodicity properties are the obvious ones. These implications are summarized in Table 3. TABLE 3
T regularly almost periodic
I
T pointwise regularly almost periodic
T regularly almost periodic
at x
T isochronous
I
I.
T weakly isochronous
T pointwise isochronous
T isochronous
uniform regular almost periodicity properties
pointwise regular almost periodicity properties
regular almost periodicity properties
at x
at a point
5.36. NOTES AND REFERENCES.
(5.04) Since T is discrete, this is a purely group-theoretic result and (2) may be found in Kurosch [1]. (5.08) Cf. P. A. Smith [1]. (5.09) This theorem and its proof are due to E. E. Floyd. (Personal communication.) (5.18) Subject to the additional hypotheses that T is discrete and abelian, this theorem was proved by Garcia and Hedlund [1]. A proof of 5.18 was obtained by S. Schwartzman [1]. (5.33) Cf. P. A. Smith [1].
6. REPLETE SEMIGROUPS 6.01. DEFINITION. A topological group T is said to be generative provided that T is abelian and is generated by some compact neighborhood of the identity element of T. 6.02. STANDING NOTATION.
Throughout this section T denotes a generative
topological group.
6.03. REMARK. The assumption that T is generative ensures the existence of "sufficiently many" replete semigroups in T. The generality of generative topological groups T is indicated by the structure theorem [Weil [1], p. 110] that T is isomorphic to C X9' X (R' where C is a compact abelian group and n, m are nonnegative integers. The theorems of this section have lemma value. Some of them do not use all of the hypothesis that T is generative. 6.04. THEOREM.
Let P be a semigroup in T. Then:
(1) If P is replete in T, then T = P-1P. (2) If T = P-1P and if T is discrete, then P is replete in T. PROOF. (1) If t E T, then there exists s E T such that st, st2 E P whence t = (st)_1(st2) E P_1P (2) Let n be a positive integer and let t1 , - , t E T. For eaeh i = 1, ,n p . Then tt; E P choose p; , q; E P such that t; = p; 1q; . Define t = pi
(i = 1,
-
,
n). The proof is completed.
6.05. REMARK.
Let P be a semigroup in (R such that P is maximal with
respect to the property of containing only positive nonintegral numbers. Then P is replete relative to the discrete topology of CR but P is not replete relative to the natural topology of (R. 6.06. THEOREM. Let P and Q be replete semigroups in T. Then PQ and P-1
are replete semigroups in T. PROOF. Obvious. 6.07. THEOREM. Let P be a replete semigroup in T and let K be a compact subset of T such that e E K. Then nkEK kP is a replete semigroup in T. PROOF. Since Q= f kEK kP = I IkER (P n kP) and each P (l kP (k E K) is a semigroup, it follows that Q is a semigroup. Let C be a compact subset of T and define D = C V K-1C. There exists t E T such that tD C P. Now k E K
implies C C D (1 kD whence tC C tD 0 kt D C P 0 kP. Thus tC C Q and the proof is completed. 6.08. THEOREM.
Let E be a subset of T such that E contains an open symmetric
neighborhood of e which generates T and let t E T. Then U.=1 t'E' is a replete semigroup in T. 57
TOPOLOGICAL DYNAMICS
58
PROOF.
[6.08)
Obvious.
The class of all replete semigroups in T has a countable
6.09. THEOREM.
of replete semigroups in T such base, that is, there exists a sequence P, , P2 , that each replete semigroup in T contains P. for some positive integer n.
PROOF. By 6.03 there exist a compact subgroup K of T and a separable ] be a countable closed subgroup S of T such that T = KS. Let [s, , s2 , dense subset of S and let U be a compact symmetric neighborhood of e whose ). Each P" interior generates T. Define P. = Ui=1 s;,U` (n = 1, 2, (n = 1, 2, ) is a replete semigroup in T. Let P be a replete semigroup in T.
There exist a neighborhood V of e and t E T such that VtKU C P. Choose k E K and s E S for which t = ks. There exists a positive integer n such that sn E Vs. Hence s.U C VsU = Vtk 'U C P and P C P. The proof is completed. Let S be a closed syndetic subgroup of T and let K be a compact
6.10. THEOREM.
subset of T. Then there exists a compact subset H of S such that K' n S C H" for all integers n.
PROOF. We may suppose that T = SK, e E K and K = K-'. Define H = K3 n S. Now H is a compact subset of S and to prove the theorem it is enough to show that K" 0 S C H", for all positive integers n. Let n be a positive integer and let k, , , k" E K such that k, . . . kn E S. If t E T, then there
exists s E S such that t E sK-' whence s E W. Thus for each integer i (1 < kn . i < n) there exists s; E S such that s; E k, k;K. Define sn = k, Clearly sn E S and sn E k, k,K. Now se's;., E Kk;+,K C K3 and s;'s;+1 E
S (1 < i < n) whence sz's;+, E H and s;+1 E s;H (1 < i < n). Also s, E k,K C K3 and s, E S whence s, E H. We conclude that k, kn = sn E sn_,H C sn_2H2 C . . C s,Hn-' C H". The proof is completed. Let S be a closed syndetic subgroup of T. Then S is a genera-
6.11. THEOREM.
tive topological group. PROOF.
Use 6.10.
6.12. REMARK.
Actually 6.11 remains true when "syndetic" is omitted.
6.13. THEOREM. Let S be a closed syndetic subgroup of T and let Q be a replete semigroup in S. Then there exists a replete semigroup P in T such that P (1 S C Q. PROOF.
Let K be a compact symmetric neighborhood of e whose interior
generates T. By 6.10 there exists a compact subset H of S such that K" (1 S C H"
for all integers n. For some s E S we have sH C Q. Define P = Un°1 s"K". Now P is a replete semigroup in T and
P n S C U (s"K" () S) C U s"H" C Q. n=1 n=1 The proof is completed.
REPLETE SEMIGROUPS
[6.191
6.14. THEOREM.
59
Let S be a closed syndetic subgroup of T and let P be a replete
semigroup in T. Then P n S is a replete semigroup in S. PROOF.
Clearly P 0 S is a semigroup. Let H be a compact subset of S
and let K be a compact subset of T for which T = SK. Since K-'H is compact,
there exists t E T such that tK-'H C P. Choose s E S and k E K such that t = sk. Now sH = tk-'H C P and sH C S. Hence sH C P (1 S and the proof is completed. 6.15. THEOREM. equivalent.
Let A C T. Then the following statements are pairwise
(1) A is extensive in T, that is, A intersects every replete semigroup in T. (2) A intersects every translate of every replete semigroup in T. (3) T = AP for every replete semigroup P in T. (4) T = AtP for every replete semigroup P in T and every t E T. PROOF. Assume (1). We prove (2). Let P be a replete semigroup in T and let t E T. By 6.07, P (1 tP is a replete semigroup in T. Hence A intersects P (1 tP and consequently tP. Assume (2). We prove (3). Let P be a replete semigroup in T and let t E T. Since A (1 tP-' 9 0, there exist a E A and p E P such that a = tp '. Thus
t=apEAP.
Clearly, (3) implies (4). Assume (4). We prove (1). Let P be a replete semigroup in T. Since T = AP-',
there exist a E A and p E P such that e = ap'. Hence a = p and A (1 P
0.
The proof is completed. 6.16. THEOREM. Let A be an extensive subset of T and let t E T. Then to and A-' are extensive in T. PROOF.
Use 6.15 and 6.06.
6.17. THEOREM.
Let A be a syndetic subset of T. Then A is extensive in T.
PROOF. Let P be a replete semigroup in T. Choose a compact subset K of T such that T = AK and then choose t E T such that tK C P. Since T =
AtK C AP, the conclusion follows from 6.15. 6.18. THEOREM.
Let A, B, K C T such that A is extensive in T, K is compact,
and A C BK. Then B is extensive in T. PROOF. Let P be a replete semigroup in T. Choose t E T such that tK C P. From 6.15 we conclude that T = AtP C BtKP C BP and that B is extensive
in T. 6.19. THEOREM.
Let S be a closed syndetic subgroup of T and let A C S.
Then A is extensive in S if and only if A is extensive in T. PROOF.
Use 6.13 and 6.14.
TOPOLOGICAL DYNAMICS
60
[6.20]
6.20. THEOREM. Let S be a closed syndetic subgroup of T, let Q be a replete semigroup in S and let K be a compact subset of T such that e E K. Then there exists a replete semigroup P in T such that PK (1 S C Q.
PROOF. By 6.13 there exists a replete semigroup R in T such that R n S C Q.
Define P= n ,,K Rk-'. By 6.07, P is a replete semigroup in T. Now PK C R whence PK n S C R n S C Q. The proof is completed. 6.21. THEOREM. Let S be a closed syndetic subgroup of T, let U be a neighborhood of e and let t E T. Then there exists a positive integer n such that t" E SU. PROOF. Let K be a compact subset of T for which T = SK. Choose a neighborhood V of e such that VV-' C U. Let a be a finite class of translates of V which covers K. To each positive integer n there correspond s E S and k E K such that t" = s k . Select positive integers p, q such that p > q and k, , k, E Vo
for some Vo E ff. Then tD The proof is completed.
= ssQ'k,kq' E Skka' C SV,,Vo' C SVV-' C SU.
6.22. THEOREM. Let S be a closed syndetic subgroup of T, let U be a neighborhood of e, let K be a compact subset of T and let k, , ka , be a sequence of elements of K. Then there exist finitely many positive integers it , , i (n > 1)
such that i, < PROOF.
subsets U1,
< in and k;,
k;, E SU.
It follows readily from 6.21 that there exist finitely many open ,
U. of T and positive integers p,
,
, p such that K C U 1
_1
U;
, m). There exists an integer j (1 < j < m) such that and U;' C S U (j = 1, k; E U; for infinitely many positive integers i. Define n = p.. Choose positive < in and k;, , , k;. E U1 . Hence integers i, , , in such that it < k;, k;w E U; C SU. The proof is completed.
6.23. STANDING NOTATION.
For the remainder of this section (X, T, r)
denotes a transformation group. 6.24. THEOREM. Let Y be a subset of X such that every replete semigroup in T contains a compact set E such that Y C YE. Then there exists a compact subset C
of T such that YT = YC. PROOF.
Let U be an open symmetric neighborhood of a such that U generates
T and U is compact. Define H = Ua and K = U3 = UH. We first show that there exists a positive integer n such that if k E K, then Y C U:-1 Yk(kH)'. To show this it is enough to prove that if ko E K, then there exists a positive integer m and a neighborhood V of e such that k E k0V implies Y C UT 1 Yk(kH)'. Now suppose ko E K. Define P = U i ko(koU)'. The set P is an open replete semigroup in T. Hence P contains a compact set E such that Y C YE. Choose a compact symmetric neighborhood V of e for which V C U and EV C P. Since EV is compact, there exists a positive integer m such that EV C U 'j-1 ko(koU)' and hence YV C YEV C U-1 Yko(ioU)'.
REPLETE SEMIGROUPS
[6.27]
61
Let k E k0V and y E Y. Choose v E V such that ko = kv. Then m
m
m
yv E U Yko(koU)` = U Ykv(kvU)' C U Ykv(kH)' i=1
i=1
i=1
and y E U-1 Yk(kH)'. This completes the proof that there exists a positive integer n such that if k E K, then Y C U;=1 Yk(kH)'. Let n denote such an integer. Choose a positive integer p (p => n) so large that if k1 , for some n + 1 of the elements k1 , , k9+1 , let us say k1 ,
kn+1 E K, then , we have k1-'k1 E U (i, j = 1, , n + 1). We now show that YT C YKD, which will complete the proof. Assume YT d YKD. Then c YK" for otherwise YT C Ui-=1 YK' C U 9 YK' C YKD. Select y E Y and kl , , k,., C K for which yk1 YKD. There exist n + 1 of the elements k1 , k9+1 , k,+l , let us say k,, , ks+1 , such that k:1k; C U (i, j = 1, , n + 1). Let r be a positive integer such that r N implies (yip, ycp,,) E a. 9.35. LEMMA. Let X be a topological space, let Y be a metric space, let (,p I n = ) be a sequence of continuous functions on X to Y which converges pointwise 1, 2,
to a function rp on X to Y and let E be the set of all x E X such that (ip I n = 1, 2,
) converges to (p uniformly at x. Then E is a residual Ga subset of X.
PROOF. Let p be the metric of Y. For each positive number a define A (e) to be the set of all x E X such that p(ycp, yyp ,) < e for all elements y of some neighborhood of x and all integers m greater than some positive integer. Clearly A(e) is open in X for all positive numbers e and E= nn-1 A(1/n). Let e be a positive number. It remains to show that B = X - A(e) is a first category subset of X. Define a = e/5. For each positive integer n define C. to be the set of all x E X such that p(xrp, xg,,.) < S for all integers m such that
m > n. By hypothesis X = Una1 C whence B = Un=1 B (1 C . Let n be a positive integer and define D = B 1 1 C . The proof will be completed when we show that D is nowhere dense in X. Assume that D is somewhere dense in X, that is, there exists a nonvacuous open subset U of X such that U C D. If x E U and if p is an integer such that p > n, then there exists an integer q such that q >_ n and p(xcp, x1p,) < 6 and there exists y E U (1 D such that p(x,p, , yrp,) < 6 and p(xcp, , y(p,)
n, then p(xp, xrp,) < e. By definition there-
fore, U C A (e) = X - B whence U (1 B = 0. However, since U C D C B and U is nonvacuous open, U (1 B
0. This is a contradiction. The proof is
completed.
be the total homeo9.36. THEOREM. Let X be a compact metric space, let morphism group of X, let G be a universally transitive subgroup of (b and let H be the centralizer of G in 4. Then H is equicontinuous. h2 , be a sequence in H. By 11.13 it is enough to show has a uniformly convergent subsequence. Choose xo c X. converges. We may suppose lim,,,, xah Some subsequence of xoh, , xoh2 , exists; call this limit yo . If x E X and if g E G such that xog = x, then yog = PROOF.
Let h1
that h, , h2 ,
,
80
TOPOLOGICAL DYNAMICS
[9.391
lim xh . Hence h, , h2 , converges pointwise to some function h on X to X. By 9.35 there exists x, E X such that h, , h2 , converges to h uniformly at x, . We show h, , h2 , converges to h uniformly at all points of X. Let x E X and let a be a positive number. There exists g E G such that x,g = x. Choose a positive number S such that z, , z2 E X with p(z, , z2) < S implies p(z,g, z2g) < e.
There exists a neighborhood U, of x, and a positive integer N such that z all integers n with n > N, where p is the metric of X. Define U = U,g. Now U is a neighborhood of x. If y E U and if n is an integer with n > N, then yg 1 E U, and p(yhg-', p(yg 'h, yg 'hn) < S
p(zh,
whence p(yh,
converges to h uniformly at x. converges uniformly to h. The proof is com-
E. This shows that h, , h2 ,
It now follows that h, , h2 , pleted.
9.37. THEOREM. Let X be a compact metric space, let T be abelian and let (X, T, r) be transitive. Then the following statements are equivalent: (1) X is an almost periodic minimal orbit-closure under (X, T, r). (2) The motion preserving group of (X, T, r) is universally transitive. PROOF.
Use 4.35, 4.38, 9.33 and 9.36.
9.38. THEOREM. Let X be a T2-space, let X be minimal under T, let rp be a (X, T)-motion preserving homeomorphism such that xxp E xT for some x E X and let T be abelian. Then (p = r` for some t E T. PROOF. Let x E X and t E T such that xv = xr'. Then xsrp = xsr' (s E T), y,p = yr` (y E xT) and yp = yrt (y E X). The proof is completed.
9.39. NOTES AND REFERENCES.
(9.02) The term transitive was used by G. D. Birkhoff in 1920 (cf. Birkhoff [1], vol. 2, p. 108 and p. 221) to denote regionally transitive as defined here. The term is commonly used with the significance of the expression universally transitive as defined here. (9.11) Cf., e.g., Kuratowski [1]. (9.12(2)) This is, in a sense, the topological analogue of metric transitivity. Cf. E. Hopf [2]. (9.17) Cf. Hilmy [2]. (9.24) Cf. Oxtoby and Ulam [2]. (9.35) Cf. Hausdorff [1], pp. 385-388. (9.36) Cf. Gottschalk.[9] and Fort [1]. Interesting examples of transitivity are to be found in Birkhoff [1], vol. 3, p. 307, Seidel and Walsh [1] and Oxtoby [1].
10. ASYMPTOTICITY 10.01. STANDING HYPOTHESIS.
In 10.02-10.10 we assume that X is a
separated uniform space, and that (X, T, 7r) is a transformation group. 10.02. DEFINITION.
Let x, y E X. The points x and y are said to be sep-
arated (each from the other) under T provided there exists an index a of X such that t E T implies (xt, yt) a. The points x and y are said to be nonseparated
(each from the other) under T provided that x 76 y and the pair x, y is not separated under T, that is, if a is an index of X, then there exists t E T such that (xt, yt) E a. Let A and B be orbits under T. The orbits A and B are said to be { separated } {nonseparated }
(each from the other) provided that A 0 B and there exist
x E A and y E B such that x and y are {separated }{nonseparated} under T. 10.03. INHERITANCE THEOREM. Let X be compact, let x, y E X and let S be a syndetic subgroup of T. Then x and y are nonseparated under S if and only if x and y are nonseparated under T. PROOF. Suppose x and y are nonseparated under T. Let a be an index of X
and let K be a compact subset of T such that T = SK. There exists an index $ of X such that (x1 , x2) E ,B and k E K implies (x,k-', x2k-') E a. Select t E T such that (xt, yt) E fl. Now select s E S and k E K such that t = sk. It follows that (xs, ys) = (xtk-', ytk-') E a. The proof is completed. 10.04. THEOREM. Let X be a compact minimal orbit-closure under T and let x, y E X such that x and y are nonseparated under T. Then the T-traces of x and y
coincide. PROOF.
Use 10.03 and 2.43.
10.05. THEOREM.
Let X be a compact minimal orbit-closure under T and
let T be regularly almost periodic at some point of X. Then the following statements are equivalent: (1) x and y are nonseparated under T. (2) The T-traces of x and y coincide.
PROOF. 'By 10.04, (1) implies (2). Assume (2). We prove (1). Let a be an index of X. Choose a symmetric index /3 of X such that 32 C a. Let T be regularly
almost periodic at z E X. There exists a closed syndetic invariant subgroup A of T such that zA C z/3. Let K be a compact subset of T such that T = AK. Since X = zAK = UkEK zkA, by 2.42 there exists k E K such that x, y E zkA whence xk-', yk-' E zA C z,6 and (xk-', yk-') E a. The proof is completed. 10.06. THEOREM. Let X be compact, let X be minimal under T and let x, y E X such that x - y and xcp = y for some (X, T)-motion preserving homeomorphism cp. Then x and y are separated under T. 81
82
TOPOLOGICAL DYNAMICS
[10.061
PROOF. Assume x and y are nonseparated. Choose open neighborhoods U, V of x, y such that U Th V = 0 and Lice C V. There exists a finite subset E of T such that X = UE. Select t E T and s E E such that xt, yt E Us whence
xts-1, yts 1 E U. It follows that yts ' =
xcpts-'
= xts 'cp E Ucp C V and
yts-' E U (1 V. This is a contradiction. The proof is completed. 10.07. THEOREM. Let X be compact, let X be minimal under T, let x, y E X such that x 3-!5 y and y E xT, and let T be abelian. Then x and y are separated under T. PROOF.
Use 9.32(4) and 10.06.
Let X be compact, let x, y E X, let both x and y be almost
10.08. THEOREM.
periodic under T, let x and y be nonseparated under T and let T be locally compact. Then neither x nor y is regularly almost periodic under T. PROOF. Assume x is regularly almost periodic under T. Then there exists a closed syndetic invariant subgroup S of T such that y Er xS. Since x and y are almost periodic under S, we have that xS (1 yS = 0. It follows from 10.03 that the pair x, y is separated under T. This contradicts the hypothesis and the
proof is completed. 10.09. REMARK.
Let X be compact, let x, y E X and let .p be a (X, T)-
motion preserving homeomorphism. Then xcp and ycp are separated under T if and only if x and y are separated under T. 10.10. THEOREM. Let X be compact, let X be minimal under T, let there exist exactly one nonseparated pair of orbits under T and let T be abelian. Then every (X, T)-motion preserving homeomorphism is a transition of (X, T). PROOF.
Let x, y E T such that x, y are nonseparated and xT 5 yT. Let
be a motion preserving homeomorphism. Since xceT, yPT are nonseparated
orbits, we have xcpT = xT or xppT = yT. It is enough by 9.38 to show that x,pT = xT. Assume xcoT = yT. Then xcpa` = y for some t E T. By 10.06, x and y are separated. This is a contradiction. The proof is completed. 10.11. STANDING NOTATION.
Throughout the remainder of this section X
denotes a compact metric space with metric p and cp denotes a homeomorphism of X onto X. It is somewhat more convenient to speak of the homeomorphism ce than the discrete flow generated by p. 10.12. DEFINITION.
Let x E X. The orbit of x, denoted 0(x), is the set
U.'-'-. x(P". The {negative} {positive} semiorbit of x, denoted {0-(x)} {0+(x)}, is the set {Un=o xcp"}{Un=0 xcp"}. To indicate dependence on P we may suffix the phrase "under cp" and adjoin a subscript cp to 0. 10.13. REMARK.
Let x E X. Then the negative semiorbit of x under cc
coincides with the positive semiorbit of x under rp '.
ASYMPTOTICITY
[10.201
83
Let x E X. Then: 0(x) = 0-(x) U 0+(x).
10.14. REMARK. (1)
0(x)c0-' = 0(xco-') = 0(x) = O(xgo) = O(x)cp. 0-(x)go
= 0-(xcp-') C 0-(x) C 0-(XP) = 0-(x)co.
0+(x),P-' = 0+(xgo ') D 0+(x) D 0+(xgP) = 0+(x),P. (2)
0(x) = 0-(x) U 0+(x).
5(x),' = 0(xg ') = 0(x) = 0(xgo) = 0(x)g. 0 (x)go '= 0-(xgo ') C 0-(x) C 0-(XP) = 0-(x)go. 0+(x),P-'= 0+(xgo ') D 0+(x) D O+(xgo) = 0+(x)v. 10.15. DEFINITION. Let x E X. The { a-limit } { w-limit } set of x, denoted {a(x) } [.(x)), is the set of ally E X such that lim1. 0. x(pn' = y for some sequence 1. Each n, , n2 , of integers such that { n, > n2 > ... } In, < n2 < point of { a (x) } { w (x) } is called an { a-limit } { w-limit } point of x. To indicate dependence on v we may suffix the phrase "under go" and adjoin a subscript gp to a and co. This definition agrees with 6.33.
Let x E X. Then the a-limit set of x under rp coincides
10.16. REMARK.
with the w-limit set of x under go '.
Let x E X. Then: 0(x) = 0(x) U a(x) U w(x). 0-(x) = 0-(x) U a(x) . 0+(x) = 0+(x) V w(x) . (2) a(x) and w(x) are nonvacuous closed invariant. 10.17. REMARK.
(1)
a(xgo ') = a(x) = a(xcp) . w( xcp') = w(x) = w(xg) . ' xr ) (4) a(x) = nn--,O w(x) = I n,-. (U (y ma-n .,,= xgm) (5) Every point of a(x) U w(x) is regionally recurrent.
(3)
(/
f
10.18. DEFINITION. Let x E X. The homeomorphism gp is said to be {negatively) { positively } recurrent at x and the point x is said to be (negatively) { posiof integers tively} recurrent under rp provided there exists a sequence n, , n2 , such that In, > n2 > . . . } In, < n2 < . . } and lim;._+ xgn' = x. The homeo-
morphism (p is said to be recurrent at x and the point x is said to be recurrent under (p provided that go is both negatively and positively recurrent at x. This definition agrees with 3.36. 10.19. REMARK.
Let x E X. Then x is negatively recurrent under rp if and
only if x is positively recurrent under go '. 10.20. REMARK.
Let x E X. Then:
(1) The following statements are pairwise equivalent: (I) x is negatively recurrent.
(II) X E a(x).
TOPOLOGICAL DYNAMICS
84
[10.201
(III) X E 0-(xP-'). (IV) 0(x) = 0(x) = a(x) D w(x). (2) The following statements are pairwise equivalent: (I) x is positively recurrent. (II) X E w(x). (III) X E 0+(x(p).
(IV) 0(x) = 0+(x) = w(x) D a(x). (3) The following statements are pairwise equivalent: (I) x is recurrent. (II) X E a(x) (1 w(x). _ (III) X E 5-(XP-I) (1 0+(x(P).
(IV) 0(x) = 0-(x) = 0+(x) = a(x) = WW. 10.21. DEFINITION.
Let x E X and let B be a closed invariant subset of X.
The point x is said to be { negatively } { positively } asymptotic to B under c provided that x ($ B and p(xo", B) = 01 {lim, .+0. p(xcp", B) = 01.
Let A be an orbit under op and let B be a closed invariant subset of X. The orbit A is said to be { negatively } { positively ) asymptotic to B provided there
exists x E A such that x is {negatively} {positively} asymptotic to B under (p. Let x, y E X. The points x and y are said to be {negatively} {positively} asymptotic (each to the other) undercp provided that x F-4 y and {lim, p(x(p", yip") = 0} { Jim,+m (xga", yip") = 0 } .
Let A and B be orbits under (p. The orbits A and B are said to be { negatively } { positively) asymptotic (each to the other) provided that A 0 B and there exist x E A and y E B such that x and y are {negatively} {positively} asymptotic under (p.
The term { asymptotic } f doubly asymptotic) means negatively for) { and } positively asymptotic. 10.22. REMARK.
Let x E X and let A be an invariant closed subset of X.
Then x is negatively asymptotic to A under (p if and only if x is positively asymptotic to A under '.
Let x, y E X. Then x and y are negatively asymptotic under o if and only if x and y are positively asymptotic under cp '. 10.23. REMARK.
Let x E X, let A be an invariant closed subset of X and
let n be an integer. Then x'p" is { negatively } { positively) asymptotic to A under cp if and only if x is {negatively} {positively} asymptotic to A under cp.
Let x, y E X and let n be an integer. Then x0" and ycp" are { negatively } { positively } asymptotic under cp if and only if x and y are { negatively) { positively} asymptotic under cp.
Let x E X. Then: (1) x is negatively asymptotic to a(x) if and only if x ($ a(x).
10.24. REMARK.
w(x). (2) x is positively asymptotic to w(x) if and only if x a(x) U w(x). (3) x is doubly asymptotic to a(x) U w(x) if and only if x
ASYMPTOTICITY
[10.30]
85
10.25. REMARK. Let A be the set of all regionally recurrent points of X. Then A is a nonvacuous closed invariant subset of X such that every point of
X - A is doubly asymptotic to A. Let n be a positive integer. Then: (1) If x E X and if A is an invariant closed subset of X, then x is {negatively} 10.26. INHERITANCE THEOREM.
{ positively) asymptotic to A under cp" if and only if x is { negatively } { positively } asymptotic to A under rp.
(2) If x, y E X, then x and y are { negatively } { positively } asymptotic under co' if and only if x and y are {negatively} {positively} asymptotic under (p. PROOF.
Obvious.
10.27. THEOREM. Let x E X and let A be a closed invariant subset of X such A. Then the following statements are pairwise equivalent: that x (1) x is {negatively} {positively} asymptotic to A.
(2) {a(x) C A}{w(x) C Al. (3) If U is a neighborhood of A, then there exists an integer n such that {
Um-n
xIpm C U} { Um-n xcpm C U}.
PROOF.
Obvious.
10.28. THEOREM. Let A be a closed non-open invariant subset of X and let there exist a neighborhood U of A such that x E U - A implies 0(x) ( U. Then there exists y E X such that y is asymptotic to A. PROOF. Let V be an open neighborhood of A such that V C U. Choose a sequence x, , x2 , of points of X - A such that lim;-.Fm x; E A. For each
positive integer i -let n; be the integer with least absolute value such that xsp ni E X - V. We may assume that 0 < n1 < n2 ... and hm,-+. xtcp"` -y E X - V. It follows that Una , y(p" C V. By 10.24(1) it is enough to show that a(y) C A. Assume there exists x E a(y) - A. Then 0(x) C a(y) C V C U. This contradicts the hypothesis. The proof is completed. 10.29. THEOREM.
Let A be a subset of X such that Ace C A and let U be an
open neighborhood of A. Then at least one of the following statements is valid:
(1) There exists a closed subset E of X such that A C E C U, EIp C E and
EC (2) There exists an open subset V of X such that A C V C U and Vv-' C V. PROOF. Define E= UIp". Now E is closed, E(p C E and E C U/p 1. Since A C Ace ' C Ac-' C , we have A C A1p" C Uco " (n = 1, 2, ) and A C E. If E r U, then (1) holds. Assume E C U. There exists a positive
integer m such that nn, c U. Define V= fln, U(p-". Now V is open and A C V C U. Since V =u (l nn,-, UP ", we have Vcp 1 =nn-1 C V. The proof is completed. 10.30. THEOREM.
Let X be self-dense and let a be a positive number. Then
there exist x, y E X with x 0 y such that n >= 1 implies p(xcp", y1p") < e.
TOPOLOGICAL DYNAMICS
86 PROOF.
[10.301
Define the homeomorphism
XXX° XXX by (x, y),p = (xp, yco) (x, y E X). The diagonal A of X X X is a closed non-open
#i-invariant subset of X X X. For n a positive integer, define U. = f (x, y) x, y E X, p(x, y) < 1/n1. Clearly U is an open subset of X X X and A C U . Now apply 10.29. If conclusion (1) holds for infinitely many positive integers n, the proof of the theorem is completed. If this is not the case, there exists a positive integer N such that n > N implies the existence of an open subset V of X X X such that A C V C U and V"(P-' C V . It follows that the set -1) = [p ' 1 is an equicontinuous set of homeomorphisms of X onto X, i = 0, 1, 2, and thus, by 11.31, 4) is totally bounded in its compact index uniformity. Let 4,+ = [,p' I i = 0, 1, 2, 1. Then 4) = 4)+ U D- is a group of unimorphisms of X onto X and it follows from 11.18 that the space-index uniformity coincides with the inverse space-index uniformity of 1. Thus the set (+ is totally bounded in its compact index uniformity, hence equicontinuous, and corresponding to
e > 0 there exists d > 0 such that x, y E X with p(x, y) < S and n > 1 imply p(xv", ycp") < E. Since X is self-dense we can determine x, y E X with x y and p(x, y) < S. The proof of the theorem is completed. 10.31. DEFINITION. The homeomorphism (p is said to be expansive provided
there exists a positive number d such that if x, y E X with x y, then there exists an integer n such that p(xcp", y(p") > d. Let x E X. The homeomorphism cp is said to be expansive at x, and the point x is said to be expansive under cp provided there exists a positive number d such that if y E X with y 54 x, then there exists an integer n such that p(xo", ySo") > d. 10.32. INHERITANCE THEOREM.
Let n be a nonzero integer. Then:
(1) cp" is expansive if and only if
,p. Let a be a closed index
of Y. There exists F E such that F X F C a,,. Then xF X xF C a (x E X),
xFXxFCa(xEX),[xgo]XxFCa(xEX),[go]XFCa,and FCgoa4'. The proof is completed. 11.04. THEOREM. Let X be a {topological} {uniform} space, let Y be a uniform space, let (D be the set of all mappings of X into Y, let 4) be provided with its space-
index topology and let 4' be the set of all { continuous } { uniformly continuous } mappings of X into Y. Then ' is a closed subset of (1). PROOF.
First reading. Let (p E 'I'. We show rp is continuous. Let xo E X
and let a be an index of Y. Choose a symmetric index,6 of Y such that ,93 C a. There exists ¢ E `I' (1 gp#,b . Select a neighborhood U of xo such that U,, C xo p,6. If x E U, then (xorp, xoi') E a, (xoiP, *) E a, (xi', x(p) E 0, (xo(p, x(p) E #3 C a and xrp C xogpa. The proof is completed. Second reading. Let (p E *. We show (p is uniformly continuous. Let ,6 be an
index of Y. Choose a symmetric index y of Y such that y3 C 6. There exists 90
FUNCTION SPACES
[11.08]
E `I' (1 SPY,
.
91
Select an index a of X such that (x1 , x2) E a implies
(xok, x4) E y. If (x, , x2) E a, then xi') E 'Y, (x,', x2#G) E 7, (x2Y', x2'p) E 7 and (x,,p, x20 E 'Y3 C (3. The proof is completed. 11.05. THEOREM.
Let X be a { topological } { uniform } space, let Y be a complete
uniform space and let - be the set of all {continuous} {uniformly continuous} mappings of X into Y. Then 4) is complete in its space-index uniformity. PROOF.
Use 11.03 and 11.04.
11.06. THEOREM.
Let X be a set, let Y be a totally bounded uniform space
and let 4) be a set of mappings of X into Y. Then the following statements are equivalent:
(1) (D is totally bounded in its space-index uniformity.
(2) If a is an index of Y, then there exists a finite partition a of X such that A E a and 'p (E 4) implies A'p X A' C a. PROOF. We show that (1) implies (2). Assume (1). Let a be an index of Y. Choose a symmetric index (3 of Y such that a3 C a. There exists a finite subset
F of 4) such that 4 _ UfEF faj, . Since Y is totally bounded, for each f E F there exists a finite partition a, of X such that A f X A f C (3 (A E a1). Define
a = ()rep ar . Clearly a is a finite partition of X such that Af X Af C a (A E a, f (E F). Let A E a and let (p E 4). We show App X A(o C a. Choose
f E F such that p E f(3. . Then x'p E x f f3 (x E A), Arp C A f (, App X Acp C A f,6 X A f,3 = 16(A f X A f ),B C ,33 C a. Hence (1) implies (2).
We show that (2) implies (1). Assume (2). Let a be an index of Y. Choose a symmetric index l3 of Y such that ,3` C a. There exists a finite partition a of X such that Aso X Acp C,6 (A E a, (p E 4)). Select a finite subset E of Y such that Y = E,B. If A E a and if p E 4), then there exists y E E such that y,3 ('1 A(p 96 0 whence Acp = y/(Acp X A(p) C y$2. That is to say, if 'p E 4), then to each A E a there corresponds at least one y E E such that A'p C y,62. Each p E 4) thus determines a nonvacuous set ,p* of mappings of a into E as follows: t E 'p* if and only if Ace C At,32 (A (E a). Since the set of all mappings of a into E is finite, there exists a finite subset F of 4) for which U,E4 rp* = UIEF f*. We show = UfEF fao . Let 'p (E 4). Select t E rp*. Now t E f* for some f E F.
Hence A'p C At,82 (A (E a) and Af C At#2 (A E a). If x E X, then x E A for some A E a whence xcp E At,2, xf E At$2 and (xf, x(p) = (xf, At) (At, x(p) E ,34 C a. Thus (xf, x') E a (x E X) and 'p E fa .. The proof is completed.
Let X be a topological space, let Y.be a uniform space and let - be a set of mappings of X into Y. If x E X, then 4, is said to be equicontinuous at x provided that if ,B is an index of Y, then there exists a neighborhood U of x such that p C 4) implies Ucp C xtp(3. The set (D is said to be equicontinuous provided that 4) is equicontinuous at every point of X. 11.07. DEFINITION.
11.08. DEFINITION. Let X, Y be uniform spaces and let 4) be a set of mappings of X into Y. The set 4) is said to be uniformly equicontinuous provided
TOPOLOGICAL DYNAMICS
92
[11.08]
that if a is an index of Y, then there exists an index a of X such that x E X and cp E 4) implies xap C x(o,3.
Let X and Y be uniform spaces and let 4' be a set of mappings of X into Y. Then: (1) If 4' is uniformly equicontinuous, then 4) is equicontinuous. (2) If X is compact and if 4' is equicontinuous, then t is uniformly equi11.09. REMARK.
continuous. 11.10. REMARK. Let X be a {topological} {uniform} space, let Y be a uniform space, let 4' be the set of all mappings of X into Y, let 4' be provided with
its space-index topology and let 4, be a {equicontinuous} {uniformly equicontinuous } set of mappings of X into Y. Then I is { equicontinuous } set of mappings of X into Y. Then NP is { equicontinuous I { uniformly equicontinuous 1. 11.11. DEFINITION. Let X be a set, let Y be a uniform space and let 4) be a set of mappings of X into Y. The set 4' is said to be bounded provided that U,E. Xcp is a totally bounded subset of Y. 11.12. THEOREM.
Let X and Y be uniform spaces and let 4' be a set of uniformly
continuous mappings of X into Y. Then: (1) If 4' is totally bounded in its space-index uniformity, then 4' is uniformly equicontinuous.
(2) If X is totally bounded, then 4' is totally bounded in its space-index uniformity if and only if 4' is uniformly equicontinuous and bounded. PROOF. (1) Let # be an index of Y. Choose a symmetric index y of Y for which y3 C t3. Since - is totally bounded in its space-index uniformity, there exists a finite subset F of P such that p E 4' implies (xf, xcp) E y (x E X) for some f E F. Select an index a of X such that (x, , x2) E a and f E F implies (x, f , x2 f) E y We show that (x, , x2) E a and *p E 4' implies (x,cp, x2c) E fl. Let (x, , x2) E a and let (p E 4'. Choose f E F such that (xf, xrp) E -y (x E X). Then (x,(o, xw) = (xAP, x, f) (x, f , x2f) (x2f, x2c) E ?'3 C a. This proves (1). (2) Assume that X is totally bounded and that 4' is uniformly equicontinuous and bounded. We show that 4' is totally bounded in its space-index uniformity. It is enough by 11.06 to show that for each index a of Y there exists a finite partition a of X such that A E a and (p E 4' implies Acp X Acp C ,3. Let ,3 be an index of Y. Choose an index a of X such that (x, , x2) E a and (p E 4' implies (xAp, x2c') E 3. Since X is totally bounded, there exists a finite partition a of
X such that A E a implies A X A C a. Hence A E a and c' E 4) implies A(p X Acp C $.
Now assume that X is totally bounded and that 4' is totally bounded in its space-index uniformity. By (1), 4) is uniformly equicontinuous. We show that 4) is bounded, that is, U,Ej) Xcp is totally bounded. Let /3 be an index of Y. Choose an index y of Y for which y2 C a. There exists a finite subset F of 4) such that (p E 4' implies (xf, xp) E y (x E X) for some f E F. Since each f E F
is uniformly continuous, UIEF Xf is totally bounded. Hence there exists a
FUNCTION SPACES
[11.171
93
finite subset E of Y such that U fE F X f C Ey. Since p E 4) implies X'P C X f y, it follows that U,E* X
4, in V. By 11.14. FF 1 (p¢ in V and 1`.f OSo in V. Let 0 be the identity mapping of X. Since o E n ff5 1 and 0 E n iF % we have So0 = 0 = 4,Sc. Hence (p, ¢ E 4), (p -3L = 41 and ff --+ 1p in 91. The proof is completed.
11.20. THEOREM.
Let X be a complete uniform space and let 4) be the group
of all unimorphisms of X onto X. Then (b is a bilaterally complete topological group in its space-index topology. PROOF.
Use 11.18 (2) and 11.19.
11.21. THEOREM.
Let X be a totally bounded uniform space and let 4) be a
symmetric set of unimorphisms of X onto X. Then 4) is totally bounded in its bilateral space-index uniformity if and only if 4) is uniformly equicontinuous. PROOF.
Use 11.18 and 11.12 (2).
11.22. THEOREM. Let X be a compact uniform space, let (P be the group of all homeomorphisms of X onto X, let 4) be provided with its space-index topology and let ! be a symmetric subset of 4). Then 1k is compact if and only if' is equicontinuous. PROOF.
Use 11.18 and 11.21.
11.23. REMARK. Let X and Y be sets, let Z be a uniform space and let
7r:X X Y->Z.Write (x,y)ir=xy(x(E X,yE Y).For xEXdefine irr: Y-+Z by yr. = xy (y E Y). For y E Y define a" : X ---) Z by x7r" = xy (x (E X). Then: (1) If X is a totally bounded uniform space and if [7r" I y E Y] is uniformly equicontinuous, then [1r I x E X] is totally bounded in its space-index uniformity. (2) If [ate I x E X] is totally bounded in its space-index uniformity, then for
each index a of Z there exists a finite partition (t of X such that A E a and y E Y implies Ay X Ay C a. 11.24. DEFINITION. Let X be a topological space, let a be the class of all compact subsets of X, let Y be a uniform space with uniformity I and let 4) be a set of mappings of X into Y. For A C X and a C Y X Y define
(A, a) o = [(gyp,') gyp, 4, E 4', (xv, xi') E a (x E A)]. I
Define
v= [(A,a),IAEa,aEI].
96
TOPOLOGICAL DYNAMICS
[11.241
It is readily verified that `U is a uniformity-base of 4). The uniformity 91 of (P generated by U is called the compact-index uniformity of 4). The topology of Y be defined by (x, p) 7r = xrp (x E X, cp E 4'). Then the compact-open topology of CF is the least
topology of - which makes a continuous.
Let 3 be the compact-open topology of 4). We show that 3 makes 7r continuous. Let x E X, let (p E - and let W be an open neighborhood of (x, cc) = xcp. There exists a compact neighborhood U of x such that Ucp C W. Define V = (U, W), . Now V is a 3-neighborhood of cc and (U X V),r C W. This shows that 3 makes a continuous. PROOF.
Now let 3o be a topology of CF which makes it continuous. We show 3 C 30 Let A be a compact subset of X, let E be an open subset of Y and let p E (A, E), . It is enough to show that U C (A, E), for some 3o-neighborhood U of V. Since (A X [cp])7r C E and A is compact, there exists a 3o-neighborhood U of cp such that (A X U)x C E whence U C (A, E), . The proof is completed. 11.44. DEFINITION.
Let X be a topological space, let a be the class of all
compact subsets of X, let 8 be the class of all open subsets of X'and let CF be a set of permutations of X. For A, E C X define
FUNCTION SPACES
111.471
101
(A, E) 1, = [p I p E 4' and Aco C E], (A, E)4* = [ c o I p E 4'
and Aco 1 C E].
Define
S_ [(A, E),,I A E Ct and E E 8],
5* = [(A,E)*bIAE(t and EE 8],
i=5l)5*. The topologies 5, Y*, 3 of fi generated by 5, 5*, 9 are called the compact-open, the inverse compact-open, the bilateral compact-open topologies of 4). It is clear
that: (1) If 4) is a group, then 3* is the image of 3 under the group inversion of 'D.
(2) 5 = 5 V 3*. 11.45. THEOREM. Let X be a uniform space and let 4' be a group of homeomorphisms of X onto X. Then the { compact-index } { inverse compact-index } { bi-
lateral compact-index) topology of 4) coincides with the { compact-open } { inverse compact-open) { bilateral compact-open) topology of 4). PROOF.
Use 11.41.
11.46. THEOREM. Let X be a locally compact T2-space and let - be a group of homeomorphisms of X. Then the bilateral compact-open topology of 4' is the least topology of 4) which makes 4) a topological homeomorphism group of X. PROOF.
Use 11.43 and 11.36.
11.47. NOTES AND REFERENCES.
The purpose of this section is to set forth those developments of the theory of function spaces needed elsewhere in the book. It can be read independently of the other sections. Most of the results can be found in Bourbaki [4], where some references to the pertinent literature can be found. See also Areas [1, 2].
PART II. THE MODELS 12. SYMBOLIC DYNAMICS 12.01. STANDING NOTATION. Let S be a finite set which contains more than one element and let S be called the symbol class. We shall use i, j, k, m, n, p, q, r, s, t as integer variables, that is variables ranging over g.
12.02. DEFINITION. A { right } { left } ray is a subset R of g such that { R = [i I p < i] } { R = [i I i 5 p] } for some p E g. An interval is a subset I of g such
that I = [i p S i < q] for some p, q E g with p < q. If p E g, then { p, +-I, { - -, p } denote the rays [i p < i], [i I i :5 p]. If p, q E g with p < q, then 1p, q J denotes the interval [i j p =< i < q]. We also write { - co, + co } = g.
12.03. DEFINITION. A { bisequence } { right sequence) {left sequence } { block } is a
function on {g } { a right ray } { a left ray } { an interval } to S.
12.04. DEFINITION. We make the following definitions: (1) If A is a { bisequence } { right sequence } { left sequence), then the reverse of A, denoted A- or A, is the { bisequence } {left sequence l {right sequence l B
such that dmn B = -dmn A and B(-i) = A(i) (i E dmn A). (2) If A is a block, then the reverse of A, denoted Av or A, is the block B such that dmn B = dmn A and B(p + q - i) = A(i) (i (E dmn A) where p, q are the first, last elements of dmn A. (3) If A is a {bisequence } { right sequence } { left sequence) { block } and if n (E g, then the n-translate of A, denoted A", is the { bisequence } { right sequence }
{left sequence) {block} B such that dmn B = n + dmn A and B(n + i) _ A(i) (i E dmn A). (4) If A and B are { bisequences } { right sequences } { left sequences } (blocks),
then A is similar to B, this statement being denoted A- B, in case there exists
n E g such that A" = B. (5) If A is a right sequence or left sequence or block and if B is a bisequence
or right sequence or left sequence or block, then A is contained in B and B contains A in case A is a restriction of B or equivalently B is an extension of A; in the event that A is a { right sequence } { left sequence } { block) we may also say that A is a { right subsequence } {left subsequence } { subblock } of B. (6) If A is a { right sequence } { left sequence } { block } and if B is a bisequence
or right sequence or left sequence or block, then A appears in B in case A is similar to a {right subsequence } (left subsequence) { subblock } of B.
(7) If A is a block, then the length of A is the cardinal of dmn A. (8) If n > 0, then an n-block is a block of length n. 12.05. DEFINITION. Let A,, , A, , be a sequence of blocks such that each A. (n > 0) is contained in A"+1 . The union of Ao , A, , , denoted 102
112.111
SYMBOLIC DYNAMICS
103
Un a A. , is the bisequence or right sequence or left sequence or block A such
that dmn A = Un=0 dmn A. and each A (n > 0) is contained in A. Variations of this notation are obvious, for example, the union of a finite class of blocks. 12.06. DEFINITION. Let A be a {right sequence) { block). Then A denotes A-° where n is the least element of dmn A. Let A, B be blocks with lengths n, m. Then AB denotes the (n + m)-block C such that the initial n-subblock of C is A and the terminal m-subblock of C
is similar to B. Analogously AB denotes the (n + m)-block D such that the terminal m-subblock of D is B and the initial n-subblock of D is similar to A. A, The meaning of AIAZ is now clear where A, , , A. are blocks except that A, may be a left sequence (if p > 1) and An may be a right sequence.
Other uses of this "indexing-by-dot" notation are obvious. For example, if , A_, , Ao , A, , are blocks, then ( A_,AoA,A2A3 . . .) denotes a certain bisequence. If x is a bisequence, then we may write x = (. . . x(- 1)i(O)x(l) . . .). If, a, b, c E S, then a denotes a certain 1-block, ab denotes a certain 2-block, abe denotes a certain 3-block, etc. 12.07. DEFINITION. Let A, B be blocks with lengths n, m. Then AB denotes the (n + m)-block C such that the initial n-subblock of C is A and the terminal m-subblock of C is similar to B. Analogously AB denotes the (n + m)-block D such that the terminal m-subblock of D is B and the initial n-subblock of D is similar to A. Other uses of this "indexing-by-roof" notation are obvious. For example, if A, A, , , An are blocks, then AIAZ denotes a certain block which contains A, .
12.08. DEFINITION. The bisequence space is the set of all bisequences, that is, the set S. We consider the symbol class S to be provided with its discrete topology and we consider the bisequence space S9 to be provided with its product topology or equivalently its point-open topology. 12.09. STANDING NOTATION.
Let X denote the bisequence space S. Let
p : X X X --), (R be the function defined by p(x, y) = (1 + sup [n j x(i) = y(i)
for IiI 0, then Define µ = QQ E X.
Q
12.29. DEFINITION.
Let A be a { bisequence } { right sequence left sequence }
{block} and let dmn A = {p, q}. The Q2-extension of A, denoted A*, is the { bisequence } fright sequence } { left sequence } { block }
such that dmn A* =
{2p, 2q + 1}, A*(2i) = A(i) (p = 0, then µ(n) = 0(n) (mod 2).
a; = 0 or 1 (i = 0, PROOF.
The statement is true for n = 0 or 1, that is, for n such that
0 -< n < 2" - 1 where p = 1. Let p > 1 and assume the statement is true for n such that 0 0 and let x be a bisequence which has a Q.-repreof x induced by (A; I i (=- g) is the unique Q"_1-representation (B; I i E g) of x such that Ao = B0B1 or B_1Bo . 12.43. DEFINITION.
sentation (A1 I i E g). The 12.44. LEMMA.
Let x E M and let n > 0. Then there exists exactly one
sentation of x.
PROOF. By 12.30(9 & 10) there exists a Qri representation of u and therefore
of ur' (i E 9). Since U:EI ua' is dense in M, it follows that x has at least one Qn representation.
For n > 0 let S. denote the statement that there exists at most one Qn representation of x. Clearly So is true. Let k > 0. Assume Sk is true. We show that Sk+1 is true. Let (A; I i E g) and (B; i (E 9) be Q,,,,-representations of x. In order to show that (A; I i E g) _ (B; i E g) it is enough to show that p = r where { p, q) = dmn A. and { r, 81 = dmn Bo . We may suppose that p 5 r p + 2k, q < s. Assume that p 5,4- r whence p < r. We seek a contradiction. If r then the Qk-representations of x induced by (A; I i (E g) and (B; i E g) are not identical which is contrary to Sk . Hence r = p + 2k. Each A; , B1 (i (E .4) is similar to Qk+1 = QkQk' or to Qx+1 = QkQk . Consideration of the common Qk-representation of x induced by (A; i E g) and (B; i (E g) now shows that x is periodic. This is a contradiction. The proof is completed. I
I
12.45. LEMMA. Let x E M and let A, B be left, right sequences such that x = AB or AB. Then:
(1) IfA-QorQ',then B.QorQ'. (2) IfBNQorQ',then A PROOF.
QorQ'.
Let x E M and let P be a left sequence such that x = PQ. By 12.34,
it is enough to show that P = Q or Q'. Let n > 0. It is enough to show that x if - 2, -1 } - Q" or Q, since the end elements of { Q" } { Qn } are { 0's } { 1's } when n is even. To show this it is enough to prove that if (A; i E g) is the Q,, representation of x, then 0 is the least element of dmn Ao . Let (A; I i E g) be the Q,, representation of x and let (B; i E g) be the Qn representation of u = QQ. Now 0 is the least element of dmn Bo . Choose y E M such that for of positive integers we have some sequence n1 , n2 , lim xor"` = lim uQ"' = Y. I
I
It follows that if 0 is not the least element of dmn Ao , then y has two different Q,, representations. The proof is completed. 12.46. DEFINITION. Let x E M, let n > 0 and let (A; I i (E g) be the Q.-representation of x. Then (A; J.i E g) is {left} {right} indexed provided that {A0 = Bi,B1} (A0 = B_1Ba} where (B; I i E I) is the Q"_1-representation of X.
SYMBOLIC DYNAMICS
[12.49)
12.47. DEFINITION.
a, = 0 or 1 (n = 0, 1,
Let a = (a0 , a, ,
109
) be a dyadic sequence, that is,
). Define the sequence F,, , F,
,
of blocks inductively
as follows: (1) F,, = ao
(2) If n > 0, then if a,+, = 0, F-11 =
FnF if
1.
Define F. = Un=o F, . Let a be a dyadic sequence. Then: (1) F. !., Q. or Qn (n > 0) in the notation of 12.28. (2) Every subblock of { F. } { µ } appears in { j } {F,,}. F. (3) If a contains infinitely many 0's and infinitely many 1's, then Fa is a bisequence such that Fa E M. 12.48. REMARK.
(4) If a contains only finitely many { 0's } { 1's 1, then { F. '-' Q or O' l { F, or Q'}.
(5) If a = (0, a, ,
) and if b = (1, a,
,
), then F. = Fb
Q
.
12.49. THEOREM. Let N be the set of all bisequences x such that x = F. or P.P. or F,Fa or P.P. or FaF, for some dyadic sequence a. Then N = M.
That N C M follows immediately from 12.48. We show M C N. Let x E M. Define the dyadic sequence a = (a0 , a, , PROOF.
)
as follows: ao = x(00).
a; = S 0T if the Q,-representation of x is {right} indexed (i > 0). We adopt the notation Fo , F, ,
of 12.47. For n > 0 let (A;
5) be the
Q,- representation, of x.
We first show that F. = Aa (n 0). Clearly F,, = Ao . Let k 0. Assume F,. = Ao . We Show F", = Ao+'. If a,.+, = 0, then Ao+' = AOAI = AoAo, _ F,F, = Fk+, . If at,, = 1, then Ao+' = Ak,Ao = Ao'Ao = FaPR = F.+, . This completes the proof that F. = Ao (n 0). We now have that F. is contained in x since F. = Un=0 F, = U.=0 Ao' and Un=o Aa is contained in x.
Case I. The sequence a contains infinitely many 0's and 1's. Since F. is a bisequence, x = F. and x E N. Case II. The sequence a contains only finitely many 0's. Then F. is a left sequence such that F. '' Q or Q'. Now x = F,P for some right sequence P. By 12.45, P Q or Q'. Therefore P P. or Pa" x = P.P. or F,FF , and x E N. Case III. The sequence a contains only finitely many 1's. Then F. is a right sequence such that F. '' Q or Q'. Now x = PF, for some left sequence P. By
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[12.49]
P. or F' , x = P.P. or P.P. , and x E N.
12.45, P - Q or Q'. Therefore P The proof is completed. 12.50. LEMMA.
Let x E M such that x does not belong to the union of the
orbits of µ, µ', v, v' and let n > 0. Then there exists m > 0 such that if (A; l i E 4) is the Qom,-representation of x, then { -n, n C dmn Ao PROOF.
Cf. the proof of 12.49.
12.51. DEFINITION.
Define r : X -* X by 0
if x(i - 1) 0 x(i)'
1
if x(i - 1) = x(i),
(xr) (i) =
(x E X, i E 9)
Define
n=I 0]. Let 8 denote the complex sphere. The transformation
iz + i -Z + 1 zG8)
(5)
is an analytic homeomorphism of 8 onto S. If µ denotes the restriction of (5) to M, then µ is an analytic isometry of J1l onto 91Z*. The following three sets coincide:
(a) The set of all geodesics parameterized by arclength in M*. (0) The set of all curves ((x(s), y(s)) I s E (R) of Class C2 in
by arclength in M* such that
y(s) z(s) - 2x(s)y(s) = 0
(s E (R),
y(s) y(s) - 2(y(s))2 + (y(s))2 = 0
(s E (R).
114
parameterized
GEODESIC FLOWS
113.051
115
(y) The union of the following four sets of curves: [((a tanh (s + c) + b, a sech (s + c)) 18 E 6i) I a, b, c E 61 & a > 01, [((a tanh (-8 + c) + b, a sech (-s + c)) I s E 61) I a, b, c E 61 & a > 0], [((a,
e8+b)
I s E (R) I a, b E a],
[((a,e 8+b) IsEE (R) Ia,bE 61].
It follows that 3tz* is complete (in the sense of Hopf and Rinow [2]) and the range of any geodesic in M* is the intersection of M with a circle in S which is orthogonal to the x-axis. Thus M is complete and the range of a geodesic in M is the intersection of M with a circle in e which is orthogonal to U. 13.03. HYPERBOLIC LINES, RAYS AND LINE SEGMENTS. Let C be a circle in S which is orthogonal to U. The set C (\ 9ii is called a hyperbolic line or h-line. Any are of C which together with its endpoints p and q lies in nt is called a hyperbolic line segment or h-line segment and is said to join p and q. Given two different points p and q of M, there exists a unique h-line segment joining p and q. Let L be an h-line and let p E L. Either of the two components of L - p together with p is called a hyperbolic ray or bray of which p is called the initial
point.
Let L be an h-line. Then L = C n nt, where C is a circle in S orthogonal to U. The two points in which C meets U will be called the points at infinity of L. Given different points u and v of U, there exists a unique h-line with u and v as its points at infinity. Let R be an h-ray. The set R has just one limit point on U and this point is called the point at infinity of R. Given p E 9TZ and u E U, there exists a unique h-ray with p as initial point and with u as point at infinity. 13.04. HYPERBOLIC DISTANCE. Let p and q be different points of 1l and let S be the unique h-line segment with endpoints p and q. All geodesic arcs
in 91z with range S have the same h-length and this h-length, denoted by D(p, q), will be called the hyperbolic distance or h-distance between p and q. The h-distance between p and q is the greatest lower bound of the h-lengths of curves of Class D' in it and joining p and q (cf. Hopf and Rinow [2]). 13.05. ISOMETRIES OF THE HYPERBOLIC PLANE. Let r be an analytic isometry of NTt onto M. Then r is a conformal (directly or indirectly) analytic homeomorphism of M onto M and r admits a unique extension r* to M U U such that r* is a homeomorphism of M U U onto M U U. Let r* have distinct fixed points u, v E U. If h denotes the hyperbolic line with points at infinity u and v, then hr = h and either all points of h are fixed under r or no point of h has this property. In the latter case it is said that r is an isometry of M onto M1 with
axis h and h is called an axis with endpoints u and v. If r is an isometry of YTt onto nz with axis h and r advances points of h toward v, v is the positive fixed point
of r and u is the negative fixed point of r. The points u and v are the only fixed points of r* in M U U. If V is any neighborhood in c of { u } { v } and A is any subset of M U U such that u, v Er A, there exists an integer N > 0 such that
In < -N, n E J} In > N, n E 9} implies A(r*)" C V.
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[13.051
To a large extent, consideration of isometries of 91L onto 911 with axes will suffice for later developments, but the collection of all isometries of 911 onto 911 admits a complete and simple analysis which we develop briefly. Let E+ denote the group of linear fractional transformations
((az+c)/(cz+a) IzES) where a, c E e with ad - cc = 1. Let y E E+. Then Ua = U, My = M, o is a directly conformal analytic homeomorphism of 8 onto S and any transformation with these properties is an element of V. It is easily verified that the restriction of a to M is an analytic isometry of M onto M.
Let a = ((az + c)/(cz + a) I z E 8) E Z+ and suppose or is not the identity. Since a + a is real, y must be either hyperbolic with fixed points on U, parabolic with fixed point on U, or elliptic with fixed points inverse to U.
Let a E E+, let a be hyperbolic, and let u, v E U be the fixed points of y. converges to Let p E S with p 5 u, v. Then the sequence p, pa 1, pv 2, converges to the one of the fixed points of a and the sequence p, py, py2, other fixed point of y. The first of these points will be called the negative fixed point of y, and the other will be called the positive fixed point of a. Let u be the negative fixed point of a, let u+ be the positive fixed point of a and let A be any
subset of S such that u , u+ ($ A. Then if V is a neighborhood of {u } {u+} there exists a positive integer N such that { n < - N, n E 411 n > N, n E 9 } implies Ay" C V. Any circular are of S with endpoints u and u+ is invariant under a. The axis of a is the hyperbolic line h with points at infinity u , u+, and these points will also be called the endpoints of the axis. Let T = a I M. Then T is an isometry of 9Th onto 911 with { negative positive } fixed point { u } { u+ } and with axis h.
Let or E E+, let T be parabolic, and let u E U be the fixed point of a. If A is any subset of S such that u EF A, and V is any neighborhood of u, then there
exists an integer N > 0 such that I n I > N, n E 9, implies Aa° C V. If C is any circle which is tangent to U at u then Ca = C. Let LT E E+ and let a be elliptic. Then y has two fixed points in S which are
inverse with respect to U. There exists a disjoint class of circles covering S except for the fixed points of a such that each member of this class is invariant under y and the fixed points of a are inverse with respect to each member of this class.
Let E- denote the set of transformations ((az + c)/(cz + d) z E 5) where I
a, c E G with ad - cc = 1. Let y E E-. Then Ua = U, Ma = M, a is an inversely conformal analytic homeomorphism of S onto S and any transforma-
tion with these properties is a member of Z-. The restriction of a to M is an analytic isometry of 911 onto M.
Let a- E E-. Then, either there exists a circle C C S orthogonal to U such that p E C implies pa = p, or a has exactly two fixed points u, v and u, v E U. In the first case y is an inversion in C. In the second case, let T be the inversion in the circle D which passes through u and v and which is orthogonal to U. Let
[13.06]
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117
Q1 = or. Then al is directly conformal, Ual = U, Mo-1 = M, a, cannot be the identity mapping, a-, (E V and uo-1 = u, vo1 = v. Thus Q1 is a hyperbolic trans-
formation with fixed points u and v. We have a- = r lv1 = rQ1 = Qtr and we conclude that a is the product of a hyperbolic transformation and an inversion in a circle orthogonal to U which passes through the fixed points of o. The transformation a is called a paddle motion. The { negative } { positive } fixed point of a- is defined to be the_{ negative } { positive) fixed point of a-, . If A is any subset of S such that u, v (Er A and V is any neighborhood of the { negative } I positive }
fixed point of v, there exists an integer N > 0 such that In < -N, n E a} In > N, n E g } implies AQ° C V. The axis of or is the axis of a, . Let r = or I M. Then r is an isometry of T onto B with axis h and with { negative } { positive } fixed point that of a-. Let E = 1- U V. Then is a group of conformal analytic homeomorphisms of S onto S and any conformal analytic homeomorphism r of S onto S such that
Ur = U and Mr = M is an element of E. Let T CM and let or = r I M. Then -a is an analytic isometry of DR onto M, and conversely, if a- is an analytic isometry
of J1l onto 3, then there exists a unique element r E E such that a = r I M. This extension of a will be denoted by 6. 13.06. HYPERBOLIC CIRCLES AND EQUIDISTANT CURVES. Since the differential metric 13.01(1) is invariant under rotations about the origin, the locus of points of Uff at constant h-distance from the origin is a circle with center at the origin 0. If p is any point of on, there exists a C E such that Oo- = p; thus the locus of points at constant h-distance r from p is again a circle C containing p in its interior, though unless p is at the origin the euclidean center of C will
not coincide with p. We call C the hyperbolic circle or h-circle with center p and with radius r. It is invariant under the group of elliptic transformations which have p and its inverse in U as fixed points. Given an h-line L which does not pass through the origin 0, there is a unique
h-line passing through 0 and orthogonal to L. By application of a suitable transformation belonging to we see that the same is true for any h-line L and point p E n, p not on L. Let q be the intersection of L and the h-line through p orthogonal to L. The h-line segment pq is the hyperbolic perpendicular from p to L and D(p, q) is the h-distance from p to L. The h-distance from p to L is denoted D(p, L). The point q is the foot of the perpendicular from p to L. By consideration of the case where p is at the origin, it is evident that the h-distance
from p to L is less than the h-length of any h-line segment joining p to any point of L other than the foot of the perpendicular from p to L. Let L be an h-line with points at infinity u and v. Let C be a circle passing through u and v with C ;-, U and C not orthogonal to U. Let A = C (1 M. Then A is invariant under every hyperbolic transformation with u and v as fixed points. Let p, q E A. There exists a linear fractional transformation a such that ua = u, va = v, pa = q. But then My = M, Ua = U, a- E V, Aa = A, and a is a hyperbolic transformation with axis L. The hyperbolic perpendicular from p to L is transformed by or into the hyperbolic perpendicular from q to
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[13.06]
L and thus all points of A are at the same h-distance d from L. We call d the h-distance from A to L. 13.07. HOROCYCLES.
Let C be a euclidean circle which is internally tangent
to U at u (E U and let H = C - u. Then H is invariant under any parabolic transformation in V which has u as fixed point and H is an orthogonal trajectory of the family of h-lines which have u as common point at infinity. We call H a horocycle and u its point at infinity.
Let H be a horocycle and let A be a circular arc such that A C H. We can parameterize A with euclidean arclength and thus define an are in with A as its range; the h-length of this arc will be called the h-length of A. Let LI and L2 be h-lines with common point at infinity u and let Hl and H, be horocycles with u as common point at infinity and such that H, is interior to H2 U u. Let { sI } { s2 } denote the h-length of the are of { Hl } { H2 } cut off by L1 and L2 and let s be the h-length of the h-line segment of L1 (or L2) cut off by
Hl and H2 . Then s2 = sle'.
(6)
To derive this formula we can assume that u is any point of U and, in particular, we can choose u = -I-1. Under the transformation 13.02(5), which is an analytic isometry of M onto !*, the image of { L, L2 } is the range of a geodesic { (x(t) = aI , y(t) = e`) t E R} { (x(t) = a2 , y(t) = e') t E R} and the image of {HI} I
I
{ H2 } is the range of the curve
{ (x(t) = t, y(t) = dI) I min (a, , a2) 5 t < max (a, , a,)) { (x(t) = t, y(t) = d2) I min (a,
,
a2) < t < max (a, , a2) } with d, > d2
We then have a,
s; = I f d; I dt
d, I
a.
I aI - a2
i = 1, 2,
and d
s
=
dy
f,l y
= log (d1/d2)
which imply (6). 13.08. ASYMPTOTIC GEODESICS IN M. Let cp be a geodesic parametrized by
arclength in 1Z and let L be the range of gyp. The sequence {(p(-1), (p(-2),
}
} converges in a to a point {u (E U} {u+ E U}. The points u , u' are the points at infinity of L. The point {u } {u+} is called the {negative) { positive } point at infinity of (p. If { V- } { V' } is an open subset of G containing { u } { u+ } then there exists { s E at } { s+ E (R } such that { s < s } {,0(1), So(2),
{s > s+} implies {p(8) E V-} {cp(s) (E V+}. Let cp and t' be geodesics parameterized by arclength in 1 let {u-, u+}
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[13.10]
119
{v , v+} be the negative, positive points at infinity of {,p} {¢} and let L = rng +'. Then (a) u = v and u+ 5-!s v+ implies lim D(,p(r), L) = 0,
r-.- m
lira D(,p(r), L) =+-.
r-.+m
(b) u 5 v and u+ = v+ implies lim D(,p(r), L) _ + -,
r-.-m
lira D(,p(r), L) = 0.
r-.+m
(c) u 0 v- and u+ 0 v+ implies lim D(,p(r), L) =+oo = lim D(V(r), L).
r-'- m
Let (p and ¢ be geodesics parameterized by arclength in 9Tt with different ranges. Then and ¢ are said to be { negatively } { positively } asymptotic provided there exists so E (R such that {lim,-_. D((p(s), &(s + so)) = 0} {lim,-+ D(,p(s),
4,(s + so)) = 01, or equivalently, provided there exists s1 E R such that {lim,._ D(,p(s + s1), ¢(s)) = 0} {lim,-+m D(.p(s + s1), k(s)) = 01. 13.09. THEOREM. Let rp and ,p be geodesics parameterized by arclength in Mt with different ranges. Then .p and ,l' are {negatively} {positively} asymptotic if and only if rp and 4, have the same { negative } { positive } point at infinity. PROOF. Let { u , u+ } { v , v+ } be the negative, positive points at infinity of {,p} {,'}. Suppose that u- F4- v-. Let d(s) = D(cp(s), rng,/). Then
lira d(s) = + o' . Let so E (R. Then D(,p(s), y'(s + so)) > d(s) and thus lim,._m D(,p(s), ,4(s + so))
+ co. It follows that 'p and 4, are not negatively asymptotic. Similarly, if u+ 5-1 v+ then 'p and ¢ are not positively asymptotic.
Now suppose u = v-. Let H be the horocycle with u as point at infinity and such that So(0) E H. Let V,(so) be the point in which H meets the range of V, and let t be the h-length of the are of H with endpoints ,p(0) and From (6) of 13.07 we have 'D(1p(s), 4'(s + so)) < tea,
s E R.
Thus lima-_0, D(,p(s), ¢(s + so)) = 0 and 'p and 4 are negatively asymptotic. Similarly, if u+ = v+ then 'p and y are positively asymptotic. The proof is completed.
13.10. THE GEODESICS FLOW OF M. Let p E M. A unitangent on 9Tt at p
is a unit contravariant vector at p. The unitangent space on 9Tt at p, denoted 0(m, p) is the set of all unitangents on 9Tt at p.
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[13.101
The unitangent space on on, denoted X, is U, 3(911, p). Let the transformation µ of X onto 911 be defined as follows. If x E X and x is a unitangent at p E t, then xµ = p. The transformation µ is the projection of X onto W. Let x1 , x2 E X, let pi = x1µ, P2 = x2µ and let &(x1 , x2) denote the absolute value of the angle between x2 and the unitangent obtained by parallel displacement of x1 to p2 along the unique geodesic segment joining pl and p2 . For r E (R+, define a, _ [(x1 , x2) I x1 , x2 E X, h(x1µ , x2µ) + 6(x1 , x2) < r]. Define ti = [a, r E (R]. It is readily verified that `U is a uniformity-base of X. Let 9.1, be the uniformity generated by V. We provide X with this uniformity and I
assign to X the topology induced by `U.. Let G = I M and let g E G. Then g is an isometry of 911 onto 911, g defines a homeomorphism dg of X onto X and the set dG = [dg I g E G] is a homeo-
morphism group of X which is universally transitive, and dG is uniformly equicontinuous relative to 9.1.. If a is an open index of X, there exists a nonvacuous open subset E of X such that for (x, y) E X we have (x, y) E a if and only if x dg, y dg E E for some g e G. Let y be the transformation of X X (R onto X defined as follows. Let x E X
and let s E a. Let be a geodesic parameterized by arclength in 911 such that x = rp(s(,) is the tangent vector to (p at cp(so). Let y = ip(s + s(,). We define (x, s)y = y. Then g _ (X, (R, y) is a transformation group of X which is called the geodesic flow of M.
Let p be a geodesic parameterized by arclength in 91t. The set of tangent vectors to (p at all elements of (R is an orbit under 9 and is denoted by 0,, . 13.11. THEOREM. Let p and 4, be geodesics parameterized by arclength in 91Z with different ranges. Then 0, and O,. are {negatively} {positively} asymptotic if and only if (p and 4, have the same {negative} {positive} point at infinity. PROOF.
The necessity follows from 13.09 and the fact that the projection
µ : X ---> 911 of X onto 911 is uniformly continuous.
To prove the sufficiency, let p and ¢ have the same positive point at infinity u. (Proof of the other case is similar.) We may suppose there exists a horocycle H(0) with u as point at infinity and such that X0(0), ¢(0) E H(0). Let H(s)
(s (=- (R) be the horocycle with u as point at infinity and such that cp(s), ,'(s) E H(s). Let v be the negative point at infinity of lp. Let a(s) (s E (R) be the hyperbolic transformation whose axis has u, v as endpoints and such that ,p(s)a(s) = cp(O). Then H(s)o(s) = H(0) and the unitangents ip(s) dc(s) _. (p(0), J'(s) du(s) are unitangents at (p(0), l(s)v(s) E H(0) which are internally orthogonal to H(0). Since or(s) is an isometry of 911 onto 911, D(lp(0), ¢(s)Q(s))
h(s) = e8h(0), where h(s) is the arclength of the are of H(s) with endpoints cp(s), ¢(s). The conclusion follows.
It is now clear that if cp and 4i are geodesics parameterized by arclength in it with different ranges such that 0, and 0,, are neither negatively nor positively asymptotic, then there exists a geodesic 0 parameterized by arclength in 911 such that Oe is negatively asymptotic to 0, and positively asymptotic to 0,,, .
GEODESIC FLOWS
113.171
121
13.12. THE HOROCYCLE FLOW. For x E X let cp. be the geodesic parameterized by arclength in on such that x = ;7s(O), let u -(x) be the negative point at
infinity of (p. and let % : (R -* 91Z be the analytic curve parameterized by arclength
in 9IZ such that rng i , is a horocycle with u as point at infinity, such that %(0) = xµ, and such that (jz(0), x) is positively oriented. Define K : X X CR -p X
such that x E X, s E (R implies that (x, s)K is the unitangent at qs(s) which is externally orthogonal to mg % . The continuous flow (X, (R, K) is called the horocycle flow of 31t and denoted by 3C. 13.13. LIMIT SET OF A SUBGROUP OF Z. Let 9 denote a subgroup of Ti. If p, q E M, then p1l 0 U = qS2 (1 U. The limit set of S2, denoted A(S2), is the set pSt (1 U where p E M. Clearly A(Q) is closed and invariant under Q.
13.14. LEMMA. Let E be a finite subset of U, let u E A(S2) and let A be an arc of U with midpoint u. Then there exists w E 0 such that Ew n (U - A) consists of at most one point. PROOF. Let (3 be the least angle formed at the origin 0 by pairs of h-rays with initial point 0 and with points at infinity distinct points of E. Since E is finite, 6 > 0. Let p E M and let a(p) be the angle subtended by A at p by h-rays with initial point p and points at infinity the endpoints of A. Then lima- a(p) = 2a. There exists a sequence w1 , w2 , of elements of 0 such that lim, Ow = u. Let N be a positive integer such that a(OWN) > 27r - 0.
Since w is conformal, (U - A)wN1 can contain at most one point of E. It follows
that EWN n (U - A) consists of at most one point. The proof is completed. 13.15. THEOREM. If crd A(S2) is finite, then crd A(S2) = 0, 1 or 2. If crd A(0) is not finite, then A(S2) is self-dense and either A(S2) = U or A(S2) is nowhere dense on U.
PROOF. We suppose that crd A(12) > 2. Let a, b, c be different points of A(U). Let u E A(S2) and let A be an are of U with midpoint u. Since w E S2 implies that aw, bw, cw E A(2), it follows from 13.14 that there exist at least two points of A(S2) in A. Thus A(S2) is self-dense.
To complete the proof it is sufficient to show that if A(S2) contains an are of U, then A(S2) = U. Assuming that the are A C A(2), we let a be its midpoint
and let S E R. There exists co E 9 such that the angle subtended by A at Ow by hyperbolic rays exceeds 2a - S. But then the angle subtended by at 0 by hyperbolic rays also exceeds 27r - S. Since Aw-1 C A(12) and S can be chosen arbitrarily small, we infer that A(S2) = U. The proof is completed. Aw-1
13.16. THEOREM. PROOF.
Let u E A(12) and let crd uSt > 2. Then uSt = A(U).
Use 13.14.
13.17. THEOREM.
Let crd A(12) > 2. Then 0 is transitive on A(S2).
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[13.171
PROOF. Let a, b E A(Q) and let A, B be open arcs of U with midpoints a, b. By 13.15, A(ct) is self-dense, and thus there exist different points u, v such that u, v E A(U) n A. According to 13.14 there exists w E 0 such that either uw E B or vw E B. Since uw, vw E A(SZ), the proof of the theorem is completed.
13.18. THEOREM.
Let crd A(S7) > 2. Then there exists at most one point of
A(Q) which is not transitive under ct. PROOF. Let u, v E A(Q) and suppose that neither u nor v is transitive under Q. It follows from 13.16 that crd uI = 1 = crd vct, or, equivalently, that utl = u, va = v, and thus, if L denotes the h-line with points at infinity u, v, then Lw = L
for every w E Q. Let p E L. It is evident that U n pcl can contain at most the two points u and v, and thus crd A(Q) 2. Then some element of ct has an axis.
Suppose that no element of cl has an axis. Let A be a closed interval
of U such that A z U and let A* = U - int A. It follows from 13.19 that if w E ct, then A*w (1 A 0 0. We show that if u E A(11) and B is a closed interval of U such that u B, then there exists w E 0 such that Bw n B = 0. Let B be such an interval and let C be a closed interval of U such that u E int C and B n C = 0. Let D be a closed interval of U such that C C int D and D (1 B = 0. Let a be a positive number less than either of the two angles subtended by h-rays at the origin 0 by the two intervals which constitute D - C. There exists E 12 such that the angle which C' = U - C subtends by h-rays at Or is less than a. But then the angle which C'r ' subtends by h-rays at 0 is less than a. Let C* = U - int C. Then the angle which C*1 ' subtends by h-rays at 0 is also less than a and since C C C CD Bw
n B = 0.
Now let u and v be distinct points of A(Sl). Let A, B be open disjoint intervals of A(Q) containing u, v respectively. It has been shown that there exists w, E ( such that A'w, C A and w2 C SZ such that B'wz C B. But then A'wlwz C B C int A', contrary to the assumption that no element of 0 has an axis. The proof is completed. 13.21. LEMMA.
Let crd A(ct) > 2. Then there exist infinitely many distinct
axes of transformations of Sl and the set of endpoints of these axes is dense in A(Sl). PROOF. It follows from 13.20 that there exists w E SZ such that co has an axis. Let L be the axis of w and let u, v be the endpoints of L. Let a E U. Then La is an axis of v 'wa, which is an element of St, and the endpoints of Lo are uo, vv. Now use 13.18.
GEODESIC FLOWS
[13.24]
13.22. DEFINITION.
The subgroup 2 of
123
is mobile provided that no point
of U is fixed under all elements of R. 13.23. LEMMA. Let 0 be a mobile subgroup of E and let crd A(S2) > 2. Then there exists a pair of axes of transformations of a such that these axes have no endpoints in common. PROOF.
Let L, with endpoints a and b, be an axis of co E Q. Not all of the
infinitely many distinct axes of transformations of 2 have a as common endpoint.
For if this were the case, since some element a E 0 moves a, there would be infinitely many distinct axes with av as endpoint, of which some one would not have a as endpoint. Thus there exists an axis L, of w, E 12 which does not have a as endpoint. Similarly, there exists an axis L2 of W2 E 52 which does not have b as endpoint.
If the statement of the lemma is not true, L, must have b as one of its endpoints, L2 must have a as one of its endpoints, and L, and L2 must have a common endpoint c. But co, moves L2 into an axis which has no common endpoint with L. The proof of the lemma is completed. Let &2 be a mobile subgroup of E with crd A(S2) > 2. Then (1) A(S2) is minimal under Q. 13.24. THEOREM.
(2) If A and B are open arcs of U such that A Cl A(R) Fd 0 0 B Cl A(S2), then there exists co E SZ such that co has an axis L with endpoints a, b such that
aEAandbEB.
(3) If 4 is the set of all geodesics
to }
{ t < t,,) implies xt ($ C. The orbit 4) under T is { positively } { negatively } transient
provided there exists x E 4) such that x is { positively } { negatively } transient. 13.29. THEOREM. Let 0 be a limit partial subgroup of E and let G = 0 1 M. Let (p be a geodesic parameterized by arclength in f such that the {negative } {positive } point at infinity of cp does not belong to A(&1). Let 0 be the projection of X onto XG . Then OO B is a {negatively) {positively) transient orbit under 9G . PROOF. Let a be a compact subset of XG . There exists a compact subset A of X such that A 0 D a. Let the { negative } { positive } point at infinity of p be u and suppose u ($ A(U). There exists a neighborhood V of u in g such that V () U,EG Ag = 0. There exists ro E R such that r E R with {r < r(,} Jr > ro} implies cp(r) E V. The conclusion follows.
13.30. THEOREM.
Let 12 be a limit-partial, mobile subgroup of
and let
G = 0 1 M. Let X* _ Uweo 0, where 4) is the set of all geodesics parameterized by arclength in Jtt such that their points at infinity belong to A(S2). Let X*G be the trace of XG on X*. Let 9*G be the restriction to Xo of 9G . Then: (1) The set of all 9a-periodic points of Xa is dense in X*G
(2) 9 is transitive. PROOF.
Use 13.24.
13.31. HOROCYCLE PARTITION FLOWS.
Let 52 be a subgroup of E+ and let
G = S2 I M. We observe that dG is a uniformly equicontinuous homeomorphism
group of X and the horocycle flow 3C is dG-orbit preserving. Define the partition XG = [x dG I x E X] of X and let XG be provided with its partition uniformity. Let 3CG = (XG , (R, KG), called the horocycle partition flow induced by G, be the partition flow on XG induced by 3C. 13.32. DEFINITION.
Let r E CR. Let x E X and let vx E 2 be the elliptic
transformation such that (xµ)v= = xµ and such that do-, rotates x through the angle r in the positive sense. The transformation (x do-, I x (E X) is an analytic homeomorphism of X onto X which we call a rotor and denote by p'.
Let ri : at --* )1t be an analytic curve parameterized by arclength in M such that mg q is a horocycle. Let f (s) = h('1(0), q(s)) (s E R). Let s E R and let (p be a geodesic parameterized by arclength in ) such that ,1(0) = p(O) and q(s) = p(r) for some r E W. Let a(s) (s E (R+) be the smallest positive angle from (r) to the unitangent externally normal to rng , at n(s). Let a(s) = -a(s) (s E (R-). Then the function a : s -> a(s) (s E Gt, s -- 0) is analytic 13.33. REMARK.
with lim,,,-m a(s) = 0, the function f : s -* f (s) (s E (R) is analytic with lim i .I-. f (s) _ + , and s E cR with s 0 0 implies p*pa(a)yf(s)pa(s) = B
(7) where { y' } {)c' }
is the s-transition of the { geodesic } { horocycle } flow in X.
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[13.341
Let S2 be a limit-entire, mobile subgroup of E+ and let
13.34. THEOREM.
G = 12 1 M. Then the horocycle partition flow induced by G is transitive. PROOF. Let 0 be the projection of X onto XG . Let (t and (B be open subsets of XG and let A = (tB-', B = (BB-'. There exists an open set C C Ap", an open set D C B with D saturated with respect to dG, and 5 E (R+ such that r E (R with I r I < S implies Cpr C Ap' and Dp' C B. By 13.27, there exists an orbit under 9G which intersects both CO and DO. Also by 13.27 the 9G periodic points of XG are dense in XG and thus there exists a 9G periodic point whose orbit
intersects both CO and DO. It follows that there exist arbitrarily large real numbers t such that Cy` n D P-4 0 and thus (cf. 13.33) we can choose s E (R 0. But then AK8 = Ap" pa(8) y r c 8)p a(8) D such that I a(s) I < S and Cyf(,' n D (Cyn') n D)p'"'> Cyr(8)pac8) and thus AK8 n B D Cyr(8)p'(') n 0, from which the conclusion follows. COROLLARY. Let S2 be a limit-entire, mobile subgroup of E+ and let G = 0 M. Then there exists a point of XG which is transitive under the horocycle partition flow.
PROOF.
The proof is similar to that of Corollary 13.27.
13.35. REMARK. Let Sl be a subgroup of E+ and let G = S2 I M. Let 0 be the projection of X onto X G and let {y'} { K' j { yG } { KG } be the s-transition of the {geodesic flow of )1t} {horocyele flow of l t} {geodesic partition flow in XG induced by G) { horocycle partition flow in X G induced by G I. Let x E X, let xO = xG a KG``, = xGK`y8 XK`ys and xGyG and let t, s E (R. Then xy'K" = GG If xG E X G is {periodic) {transitive} under the horocycle partition flow
3CG and r E CR, then EG7G is { periodic } [transitive) under 3CG. 13.36. LEMMA.
Let 12 be a limit-entire, mobile subgroup of E+ and let G = Sl M.
be the t-transition of the horocycle partition flow in XG induced by G. Let a C XG be the orbit of a periodic point of the geodesic partition flow aG and let
Let KG'
B = U,, 6, aKa . Then 63 = XG . PROOF. Let x AB-'. Let (p be the geodesic parameterized by arclength in M such that x = (O). The range of (p is the axis of some element of G and let u be the negative point at infinity of (p. Let B be the set of unitangents of X which are externally orthogonal to the horocycles with u as point at infinity. Then (B = BO. Since, by 13.24(1), uSt = U, the set B dG is dense in X and since (B = BO = (B dG)O it follows that (B is dense in XG . The proof is completed. 13.37. THEOREM.
Let S2 be a limit-entire, mobile subgroup of E+ and let
G = SZ I M. Let a E XG and let a be a periodic point of the geodesic partition flow Then a is a transitive point of the horocycle partition flow 3CG .
c7G .
PROOF. It follows from 13.34(Corollary) that we can choose b E XG such that b is transitive under the horocycle partition flow 3CG . Then (cf. 13.35)
s E (R implies that by' is transitive under 3CG . Let p E (R+ be such that aya = a and let 3) be an open subset of XG . There exists r E (R+ such that s E (R with
GEODESIC FLOWS
[13.391
127
-2p - so implies Ap'pa(8)yfc8)p8(8) (1 Dpa 8) D (C'/ n D)pa(8> 0 0, from which AK' (1 B D
the conclusion of the theorem follows. 13.41. COMPLETE TWO-DIMENSIONAL RIEMANNIAN MANIFOLDS OF CONSTANT
Let 91 be a complete two-dimensional analytic Riemannian manifold of constant negative curvature -1. There exists (cf. H. Hopf [1]) a group G of isometries of AIL onto 1 such that nl is the universal covering manifold of 9'G with G the covering group. The two-dimensional Riemannian manifold MG obtained by partitioning 9 by G is isometric to R. NEGATIVE CURVATURE.
The group G has the property that p E ) implies the existence of a neighborhood W of p such that pg E W, g E G, only if g is the identity mapping. A group of isometrics of 1 with this property will be said to be discrete in M. Let F be a group of isometrics of M onto 91Z which is discrete in M. The two-dimensional Riemannian manifold obtained by partitioning JCL by F is then a complete two-dimensional analytic Riemannian manifold of constant negative curvature -1. Thus the problem of constructing the class of such manifolds is equivalent to the problem of constructing the class of groups of isometrics of 9tt which are discrete in M. 13.42. LEMMA.
Let SZ be a subgroup of
with crd A(12) > 2 and let G = 12 1 M
be discrete in M. Then Sl is mobile.
PROOF. We assume that 12 is not mobile and thus that there exists u E U such that uSl = u. Let wo E 9 be the identity mapping. Since G is discrete in f, w E 0, co 0 wo , implies that co is either parabolic, hyperbolic or a paddle motion. If all co E 0 other than co. were parabolic, the set A(11) would consist of [u], contrary to hypothesis. Thus there exists co, E Sl with axis Ll and L, must have u as one of its endpoints. If L, were the axis of every member of St other than co. ,
it would follow that A(12) = 2, which is not the case. Let co E 0 such that L,
[13.44]
GEODESIC FLOWS
129
is not the axis of w. If w does not have an axis, then w is parabolic and Llw is an axis of w-'wiw E 0 with Llw 0 L, . Thus there exists w2 E Sl with axis L2 such that L, 0 L2. Since u12 = u, Ll and L2 must have u as common endpoint.
Let p1 E L, , let p2 E L2 and let S E W. There exists m E g such that 0 < D(plwi , L2) < S and thus there exists n E 9 such that the h-distance from p,w;w2 to some point of the h-line segment joining p2 and p2(02 is positive and less than S. Since a can be chosen arbitrarily small there exists a sequence {wk I k E 9', wk E Sl} such that p,w? 0 p,w* , i F& j, and such that the sequence {p,w,k* I k E g+} converges to a point p E L2 . But this implies that G is not discrete in Mt. The proof of the lemma is completed. 13.43. THEOREM. Let 12 be a subgroup of M, let G = 12 I M and let G be discrete in M. Let wo be the identity mapping of G. Then exactly one of the following statements is valid. (1) crd A(0) = 0;12 = [wo]. (2) crd A(12) = 1; there exists w E 12 such that w is parabolic with fixed point A(12) and 12 = { w" I n E .41. (3) crd A(12) = 2; there exists w E 12 such that w is hyperbolic with fixed points
A(12)and12 = {w"InE9}. (4) crd A(12) = 2; there exists w E 9 such that co is a paddle motion with fixed points A(12) and Sl = { co" I n E 4 1.
(5) crd A(12) > 2;12 is mobile and limit-partial. (6) crd A(12) > 2; Sl is mobile and limit-entire. PROOF. Since G is discrete in 91l, co E 12 with co F& coo implies that w is parabolic, hyperbolic or a paddle motion. Thus if A(12) = 0, it follows that 12 = [wo]. Suppose crd A(12) = 1. Let u = A(12). Then w E SZ with w 0 wo implies that
co is parabolic with fixed point u. Let p E 59 and let H be the horocycle with point at infinity u and such that p E H. Then p1l C H and since G is discrete in ni, there exists p, E $2 such that p' 0 p and h(p, , p) _ 2, it follows from 13.42 that 12 is mobile. It follows from 13.15 that all possible cases have been considered. The proof is completed. 13.44. REmARc. Let 12 be a subgroup of X, let G = 12 I M and let G be discrete in M. Then G is countable and if p E M, co, , w2 E 0, pwl = pw2 then co, = w2 .
TOPOLOGICAL DYNAMICS
130
[13.451
13.45. DEFINITION. Let 12 be a subgroup of Z, let G = 12 1 M and let G be ], where discrete in M. Let po be the origin 0 and let p0G = [p; I i = 0, 1, 2, h(p, p.)]. p; 0 pi provided i 0 j. For n E 9+ define R. = [p I p E t, h(p, The set R= (fin ,R is the fundamental region of G.
Let 0 be a subgroup of E, let G = 0 1 M, let G be discrete in 91t and let R be the fundamental region of G. Then: (1) R is a non-vacuous open subset of M. (2) R is h-convex in the sense that p, q E R implies that the h-line segment joining p and q lies in R. (3) If g, , g2 E G and Rg, n Rg2 0 0 then g, = 92 . (4) Corresponding to any compact subset A of on there exists a finite subset E of G such that A_C U,,E Rg. (5) a,,(R) = ah(R) 13.46. REMARK.
13.47. THEOREM.
Let 12 be a subgroup of Z, let G = 12 1 M, let G be discrete
in n and let R be the fundamental region of G. If R is of finite h-area, then crd A(0) > 2 and 0 is mobile and limit-entire. PROOF. It follows from 13.43 that if crd A(0) 5 2 then R is not of finite h-area. Thus we can assume that crd A(0) > 2 and by 13.42 12 is mobile. We show that 12 is limit-entire. Let 5 E W. There exists r E 61, 0 < r < 1, such that if C, denotes the circle with center 0 and euclidean radius r, then any h-convex subset of 59 which is exterior to C, has euclidean diameter less than 5. There exists a finite subset E of G such that C, and its interior are contained in U,EE Rg. Let u E U and let W be a euclidean neighborhood of u of diameter less than 5 and exterior to C, . Then W (\ f is of infinite h-area and there must exist g* E G, g* ($ E, such that Rg* n W 0 0. But then Rg* is exterior to C, , the euclidean diameter of Rg* is less than 5 and Og* is within euclidean distance 25 of u. It follows that u E A(0) and thus U C A(12) and 12 is limit-entire.
13.48. GEODESIC FLOWS OF TWO-DIMENSIONAL MANIFOLDS OF CONSTANT
Let t be a complete two-dimensional analytic Riemannian manifold of constant curvature -1. Let p E 91. A unitangent on 9L at p is a unit contravariant vector at p. The unitangent space on t at p, denoted 5(9l, p) is the set of all unitangents on Yt at p. The unitangent space on 91, denoted X, is U,Ea 3(gt, p). Let x E X and let NEGATIVE CURVATURE.
x be a unitangent at p E X. Let r E 6t+ and let A, (x) be the set of all unitangents
on 9Z at p and forming an angle less than r with x. Let U,(x) be the set of all unitangents obtained from A, by parallel transport along all geodesic segments of length less than or equal to r and with initial point p. For r E 6t+ define a, = [(x, , x2) I x, , x2 E X, x2 E U.(xl)] Define `U = [a, I r E 6t+]. It is readily verified that U is a uniformity base. Let 9t be the uniformity generated by `U. We provide X with this uniformity and assign to X the topology induced by 91. Let y be the transformation of X X 6t onto X defined as follows. Let x E X
GEODESIC FLOWS
[13.52]
131
and let s E a. Let p be a geodesic parameterized by arclength in 91 such that x = (so) is the tangent vector to ip at p(so). Let y = cp(s + s(,). We define (x, s)y = y. Then G = (X, (R, -y) is a transformation group on X which is called the geodesic flow of 91. 13.49. THEOREM. Let 91 be a complete two-dimensional analytic Riemannian manifold of constant curvature -1 and of finite area. Then the geodesic flow of
91 is regionally transitive, regionally mixing, and the periodic orbits of the geodesic flow of 9Z are dense in the space of unitangents on 9Z. PROOF.
Use 13.27, 13.39 and 13.47.
13.50. CONSTRUCTION OF TWO-DIMENSIONAL MANIFOLDS OF CONSTANT NEGA-
As indicated in 13.41, the problem of construction of twodimensional Riemannian manifolds of constant negative curvature -1 is equivalent to the problem of construction of groups of isometries of M1 which are discrete in M. The problem can be completely solved by geometric methods TIVE CURVATURE.
involving the construction of fundamental regions (cf. Fricke-Klein [1], Koebe [1] and Lobell [1]). These manifolds include compact orientable manifolds of genus
at least 2 and compact non-orientable of every topological type other than the projective plane and Klein bottle. 13.51. n-DIMENSIONAL MANIFOLDS OF CONSTANT NEGATIVE
CURVATURE,
n > 2. A large number of the results of this section can be extended to manifolds of constant negative curvature of dimension exceeding 2 and the proofs have been so designed that these extensions obtain with scarcely any modifications of the proofs given for the case of dimension 2. In particular, the extensions
of the results concerning the density of the periodic geodesics and regional transitivity of the geodesic flow are valid. The extension of the concept of the horocycle flow is not immediately obvious, but mixing properties can be attained for higher dimensional manifolds (cf. E. Hopf [3]). The construction and classification of manifolds of constant negative curvature and dimension exceeding 2 is largely an unsolved problem. Compact manifolds of constant negative curvature and of dimension 3 have been constructed by Lobell [2] and Salenius [1], while non-compact manifolds of finite volume are known to exist, but these examples appear to represent only a small number of the possibilities. 13.52. NOTES AND REFERENCES.
(13.01) This model of the hyperbolic plane is commonly associated with Poincare due to his extensive use of it in the development of the theory of automorphic functions, although it appears to have been known earlier to Beltrami (cf. Beltrami [1]). (13.02) Cf. Bianchi [1], p. 584. (13.11) Cf. E. Hopf [3], p. 268.
(13.13) It is usually assumed in the definition and analysis of limit sets that G = 0 1 M is properly discontinuous (cf. L. R. Ford [1]).
132
TOPOLOGICAL DYNAMICS
[13.52]
(13.27, 13.34, 13.39) Cf. Hedlund [2] for references to the literature. (13.41) The expression discrete in M replaces the more commonly used expression properly discontinuous. The phrase discrete and without fixed points in T would, perhaps, be more appropriate. A group of isometries of can be topologized in various ways. It can be considered as a set of mappings of t
onto Yl and assigned the compact-open topology. It can be considered as a set of mappings of X onto X and assigned the compact-open topology. It can be considered as the restriction of a subgroup of 7, and thus be topologized by defining a base a for the neighborhoods of the identity of E as follows: let e E 6t+,
let
[as +c a,cES;ad -cc= 1;1a-11 <e,IcI <eT cz -+d and let (B = UeER+ B(e). Then G is discrete in M, as defined in 13.41, if and only if no element of G other than the identity has a fixed point in 1, and G is discrete in each of the stated topologies.
14. CYLINDER FLOWS AND A PLANAR FLOW 14.01. STANDING NOTATION. Throughout this section Y denotes a topological space, aR denotes the set of real numbers with the natural topology, F(Y) denotes
the set of all continuous functions on Y to at and H(Y) denotes the set of all homeomorphisms of Y onto Y.
Let X = Y X aR and let v denote the projection of X onto aR defined by (y, r)v = r (y E Y, r E (R). The subset A of X is {bounded 14.02. DEFINITION.
above) { bounded below } provided AP is { bounded above) { bounded below}, and
A is bounded provided A is bounded both above and below. 14.03. DEFINITION.
Let 0 E H(Y) and let f E F(Y). Let X = Y X aR
and let So be the homeomorphism of X onto X defined by (y, r)cp = (yO, r + f (y))
(y E Y, r E a). The homeomorphism p will be denoted q(Y, f, 0) and called the cylinder homeomorphism determined by Y, f and 0. 14.04. DEFINITION. Let X = Y X aR, let s E aR and let X -* X be defined by (y, r)0. = (y, r + s) (y E Y, r E (R). The homeomorphismk. of
X onto X will be called the translation of X by s.
14.05. REMARK. Let X = Y X aR, let s E aR and let ¢, be the translation of X by s. Let f E F(Y), let 0 E H(Y) and let (p = po(Y, F, 0). Let x E X, let fi(x) be the orbit of x under c and let r(x) = c1(x). Then (1) (2) (3)
x4,.,P.
'(x)4,. _ (k(x ¢.) .
r(x)V,. = r(x4,.)
14.06. DEFINITION. Let f E F(Y), let 0 E H(Y) and let (p = cp(Y, f, 0). Let { A- } { A+ } { SZ- } { Sl+ } denote the set of a1 lx E X = Y X aR such that {the
negative semiorbit of x under (p is bounded below) {the negative semiorbit of x under o is bounded above } { the positive semiorbit of x under 'p is bounded below the positive semiorbit of x under (p is bounded above 1. Let { a } { a+ } { w } { w+ } denote the set of all x E X such that
{ lim xcpnv = - co } { lim WP = + - I I lim xco v = - c } { lim x5onv n-.-m
n-.-m
-++m
Let B+ = A+ (\
n- +m
W. Let B- = A- (1 Q. The collection of sets [A-, a, a+, w, w+, A-', A+', fl-', St+', B+, B-] will be denoted by D(,p).
=+ A+,
}
.
Q-, SZ+,
14.07. REMARK. Let 0 E H(Y), let Y be a minimal orbit-closure under 0, let f E F(Y) and let e = cp(Y, f, 0). Let s E aR and let 4,, be the translation of X by s. Let D E D (cp) . Then D is invariant under p, D is invariant under and if D 76 0 then D is dense in X. 133
TOPOLOGICAL DYNAMICS
134
14.08. LEMMA.
[14.081
Let Y be a compact metric space, let 0 E H(Y), let f E F(Y),
let (p = (p(Y, f, 0) and let v be the projection of X = Y X (R onto (R. For each i E 9+ let there exist x; E X and n; E 9+ such that { x,(p''v - x,v > i } { x,(P "v - x, v < Then (1)
{A+ 0 0} {A- 0 0}.
(2)
{12-00} {12'00}.
PROOF. It is sufficient to prove the first reading. We assume that corresponding to i E 9+ there exists x; E X and n, E 9+
such that x,(p"v - x,v > i. Corresponding to i E 9+ let p, , q, E 9+ be so chosen (x,p'v - x,(p"v). that 0 i and k E 4, p, _5 k i} {x,(p"1v - x,v < -i) and {x,(p"1v - x{SO'`V > i} {x,(p"'V - x,(p''v < -i).
Then {B+ 0 0}{B- 0 0}. PROOF.
It is sufficient to prove the first reading.
We assume that for each i E 9+ there exists x, E X and m; , n, E 9+ such
that 0 < m, < n, and x,co"v - x,v > i, x,(p"P - x,(p"v > i. Let p, E 9+ be such that 0 < p, i and x,(p9"v - x,(p"v > i, i E 4+, in consequence of which lim,.+m p, = +
and lim,-+m (n, - p,) = + -. Let s, = x,(p"v, let 4,_ be the translation of X by -s, and let x* = x,(p"','_s,, i E 9+. Then x*v = 0 and x*(pkv = x,(P,,i x,SoD'+kv
- p, + 1,
-
- x,(p?'v, i E 9+, k E 9, and consequently x*(pkv 0, then w+ = X = a- and if f Y f(y) dµ(y) < 0, then w- = X = at In either case (6) is not valid and we infer that f f (y) dµ(y) = 0. Since x = (y, r) E X and n E J+ implies x,pn = (yO', ED-o f (yB")), it follows from (6) that E (y0P) is not bounded on I' X g+. The proof that (6) implies (7) is completed. Assume (7). We prove (5). Since f y f (y) dµ(y) = 0, it follows that
f
Y
a=0
f(yep)
dµ(y) = 0,
nE
g+,
and thus, corresponding ton E y+ there exists yn E Y such that EpI .f (ynO") = 0. Let xn = (y , 0). Then xv = 0 and xnlpnv = 0. Corresponding to m E 9+ there , n(m), satisfy exists n(m) E y+ such that not all the points xn(,n) ', j = 0, 1, the condition I xv I 5 m. For otherwise there would exist a bounded orbit under
gyp, and from 14.11 it would follow that En o f (yO') is bounded on Y X 0. If B+ 0 and contrary to hypothesis. By 14.09, either B+ 5 0 or Bx E B+, it follows from 14.11 that x E A-' f T' and thus (5) is valid. If B- F-4 0 and x E B-, it follows from 14.11 that x E A+' (l Sl+' and thus (6) is valid. But it has been shown that (6) implies (5), and thus, in either case, (5) is valid. The proof of the theorem is completed. 14.14. EXAMPLES OF CYLINDER FLOWS. It follows from 14.11 that it is easy to construct nontrivial examples of cylinder homeomorphisms which are point-
wise almost periodic. Using the notation of 14.11, we choose g E F(Y) and define f E F(Y) by f (y) = g(yO) - g(y), y E Y. Then (p = cp(Y, f, 0) is pointwise almost periodic.
It is more difficult to construct examples of transitive cylinder flows. The following method yields such examples. Let Y be a compact connected separable abelian (additive) topological group. Then Y is monothetic (cf. Halmos and Samelson [1]). Let y* be a generator of Y;
that is, Y = [ny* I n E .4]-. Let 0 : Y - Y be defined by yO = y + y*, y E
Y.
Then Y is an almost periodic minimal orbit-closure under 0. Let C be the unit circle zz = 1 of the complex plane Z. Let X : Y -p C be a character of Y such that x(y*) = e'' 0 1. Since Y is connected, ,li/a is irrational. Let 0 < nl < n2 < . . . be a sequence of integers such that E.'-, I x(nky*) is convergent. Since 0/7r is irrational, such a sequence 1 I = Ek=, I e'n`# - 1 exists. Let (an I n E 4) be defined by:
-
I
J an = l
0 unless n E [nk I k E I+] or n E [-nk I k E g+],
l an = a_.,. = I
e'nko
-1
I,
k c g+
The series E±m an x(ny) is absolutely and uniformly convergent on Y. Le f : Y --* Z be defined by f (y) = E±.' anx(ny), y E Y. Then f is continuous on Y and since x(-y) = x(y), y E Y, it follows that f (y) E lR, y E Y. By the
CYLINDER FLOWS AND A PLANAR FLOW
[14.18]
139
orthogonality property of characters, n E 5, n 5;6 0, implies f Y X(ny) dµ(y) = 0, where µ is the normalized Haar measure on Y, and thus f y f (y) dµ (y) = 0. Now suppose that there exists g E F(Y) such that
y E Y.
.f(y) = g(y + y*) - g(y),
(A)
Let b = f r g(y)X(ny) dµ(y), n E y. Then bkbk < -. But a simple - 1), n E 4. Thus computation shows that an = bn(e;nfl
"nk =
I
e" - 1
a.,
k E 9+,
1,
We infer that there cannot exist g E F(Y) such that
and hence E±m bkbk (A) is valid.
Let Y be also locally connected and let p = cp(Y, f, 0). It follows from 14.11 that En-o f (y0D) is not bounded on Y X g+ and thus, by 14.13, the discrete flow generated by 'p is transitive. 14.15. REMARK. The dyadic tree is a dendrite whose endpoints form a Cantor
discontinuum and whose branch points are all of order three. There exists a homeomorphism 'p of the dyadic tree X onto X such that: (1) p is regularly almost periodic on X. (2) cp is periodic at every cut point of X. (3) The set of all endpoints of X is a minimal orbit-closure under 'p. 14.16. REMARK. The remainder of this section is devoted to the construction of a compact, connected plane set which is minimal under a homeomorphism and which is locally connected at some points and not locally connected at other points. 14.17. REMARK.
Let f be a continuous real-valued function on a dense
subset of a real interval such that the closure of (the graph of) f in the plane is compact, connected and locally connected. Then f is uniformly continuous. (Since f is compact, it is enough to show that the relation f is single-valued. This may be done by use of the arcwise connectedness theorem.) 14.18. DEFINITION.
Let A. = [2k7r I k E J], let Xo = '31 - Ao and let
fo : Xo -* (R be defined by: fo(x) = sin 7r2
IX
-7r
x G 7r,
x 54 0,
J iII
fo(x+2ir) = fo(x),
xEX0.
For n E 1, n 0 0, let An = [2k7r + n I k E 5], let X. = (R - A. and let
(
fn : Xn -* (R be defined by f n(x)= f .(x - n), x E Xn .
Let X = (R-
UnE9 A. =
I
InES
f(x) = 10 +
Xn and let f : X ---> (R be defined by:
2-'n'fn(x),
x E X.
140
TOPOLOGICAL DYNAMICS
[14.191
14.19. REMARK. We adopt the notation of 14.18. The following statements are valid: (1) f is continuous on X.
(2) If I is any open interval of (R, then f is not uniformly continuous on
X(lI.
(3) f is 5-chained for every 8 E (R+ 14.20. LEMMA. We adopt the notation of 14.18. Let N = [(x, f (x)) I x E X]. We consider N as a subspace of the product space (R X R. Let ¢ : N N be defined by (x, f (x))+/' = (x + 1, f (x + 1)), x E X. Then ¢ and -' are uniformly continuous homeomorphisms of N onto N. PROOF. It is clear that ¢ is a one-to-one transformation of N onto N. That ¢ and ¢-' are continuous on N follows from the continuity of f on X. Thus ¢ is a homeomorphism of N onto N. We prove that ¢ is uniformly continuous on N; the proof that -' is uniformly continuous is similar. Suppose that k is not uniformly continuous on N. Then there exists e E 6i.+ such that corresponding to 8 E (R+ there exist x, x' E X such that I x - x' I < 8, I f (x) - f (x') I < 8 and I f (x + 1) - f (x' + 1) I > e. Let (Sn n E 9+) be a sequence of positive real numbers such that 81 > 82 > ... and limn-+m 8n = 0. Then there exists a sequence of pairs ((xn , xn) I n E 9+) such that x , xn E X, I X. - xn I < B. , I f (xn) - f (X.) i < Sn , f (xn + 1) - f (x.' + 1) I > e, for
all nE9+ We can assume that lim-+m xn = x = lim., x; , where x E R. Since lim infra-+m I f (xn + 1) - f (xn + 1) I > e it follows that x + 1 Er X and there
exist k, m E 9 such that x + 1 = 2k7r + m. If n E 9+, then e < I f(xn + 1) - f(xn + 1)
I=IE
2-'
[fn(xn
+ 1) - fn(xn + 1)]
There exists M E 9+ such that M > m and 12-"'[fv(xn + 1) - fn(xn + 1)]
< e/3
and
E 2-'ni[f9(xn + 1) - fn(xn + 1)] M+1
<e/3,
nE9+
E 2 -'P' [fp(Xn + 1) - f,(xn + 1)] > e/3, -M
n E 9+
Thus M
Since p E 9, p F-4 m, implies lim I fn(xn +
1)
- fn(xn + 1) 1 = 0,
CYLINDER FLOWS AND A PLANAR FLOW
(14.23]
there exists P E 9+ such that n E 9+, n > P. implies UU
provided n > P, n E
12-Iml[ fm(xn + 1)
{{=fm-1(xn), n E 9+, we have
f m(x'. + 1)] 1 > e/4. Since fm(xn + 1) 2-Im1[fm-1(xn)
141
- m-1(xn)] I > e/4
4+.
Since n E 9+ implies +m I
E 2- l" [fa(xn) - fn(x:) ]
f(xn) - f(x;) I =
< un ,
there exists Q E 9+, Q > m, such that Q`
2-Ivl[ f (xn)
- fn(xn)]
< Sn + e/10
(n E 9+).
Since p E 9, p 0 m - 1, implies lim n-++m
I
,(xn) - f y(xn) I = 0,
there exists S E 9+ such that n > S implies 12-1--11[fm-1(xn)
- fm-1(x.')] I < Sn + e/9.
Let t E 9+ with t > P such that 12-Im-11[fm-1(xt)
- f'n-1(x!)]
I
tl-Im-ll I fm-1(xt) 4 = 8. I > fm-1(xt) g 2-1-1
From this contradiction we infer the validity of the lemma.
14.21. DEFINITION. We adopt the notation of 14.18. Let M be the graph of r = f (0), 0 E X, in polar coordinates. Let tp : M -+ M be defined by (0, f (0))tp =
(0 + 1, f (0 + 1)), 0 E X. 14.22. THEOREM. The transformation p : M M is a uniformly continuous homeomorphism of M onto M, tp 1 is uniformly continuous on M, tp and p-' are pointwise almost periodic, and M is a minimal orbit-closure under (o. PROOF.
It is obvious that c is a homeomorphism of M onto M, that tp and
are pointwise almost periodic and that M is a minimal orbit-closure under tp. The uniform continuity of tp and tp 1 follows from 14.20. (P-1
14.23. REMARK.
Let X be a compact metric space, let Y be a nonvacuous
subset of X and let (p be a pointwise almost periodic homeomorphism of Y onto Y such that (p and tp 1 are uniformly continuous on Y and such that Y is a minimal orbit-closure under (p. Then there exists a homeomorphism of Y onto Y such that I Y = (p and 7 is a minimal orbit-closure under gyp.
TOPOLOGICAL DYNAMICS
142
[14.24]
14.24. THEOREM. Let M be the plane set defined in 14.21. Then M is compact, connected, and locally connected at some points but not locally connected at other points. There exists a homeomorphism 77 of M onto M such that k is a minimal orbit-closure under ,. PROOF.
Clearly M is locally connected at each of the points (0, f (0)),
0 E X. To complete the proof, use 14.17, 14.19(3), 14.20, 14.22, 14.23. 14.25. NOTES AND REFERENCES.
(14.01) Cylinder homeomorphisms with Y a circle were considered by A. S. Besicovitch [1, 2], who constructed transitive models of cylinder flows. (14.15) Cf. Zippin [1], pp. 196-197 and Gottschalk [4]. (14.16-14.24) This example of a minimal set is due to F. B. Jones (Personal communication).
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144 CAMERON, R. H.
1. Almost periodic transformations, Transactions of the American Mathematical Society, vol. 36 (1934), pp. 276-291. CARATH19ODORY, C.
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1. Automorphic Functions, New York, 1929. FORT, M. K., Jr. 1. A note on equicontinuity, Bulletin of the American Mathematical Society, vol. 55 (1949), pp. 1098-1100. FRANKLIN, PHILIP
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1. (With F. Klein) Automorphe Functionen, vol. I, Leipzig, 1897. GARCIA, MARIANO
1. (With G. A. Hedlund) The structure of minimal sets, Bulletin of the American Mathematical Society, vol. 54 (1948), pp. 954-964. GorrscHALK, W. H. 1. An investigation of continuous mappings with almost periodic properties, Dissertation, University of Virginia, 1944. 2. Powers of homeomorphisms with almost periodic properties, Bulletin of the American Mathematical Society, vol. 50 (1944), pp. 222-227. 3. Orbit-closure decompositions and almost periodic properties, Bulletin of the American Mathematical Society, vol. 50 (1944), pp. 915-919. 4. A note on pointwise nonwandering transformations, Bulletin of the American Mathematical Society, vol. 52 (1946), pp. 488-489. 5. (With G. A. Hedlund) Recursive properties of transformation groups, Bulletin of the American Mathematical Society, vol. 52 (1946), pp. 637-641. 6. Almost periodic points with respect to transformation semi-groups, Annals of Mathematics, vol. 47 (1946), pp. 762-766. 7. Almost periodicity, equi-continuity and total boundedness, Bulletin of the American Mathematical Society, vol. 52 (1946), pp. 633-636. 8. Recursive properties of transformation groups II, Bulletin of the American Mathematical Society, vol. 54 (1948), pp. 381-383.
9. Transitivity and equicontinuity, Bulletin of the American Mathematical Society, vol. 54 (1948), pp. 982-984. 10. (With G. A. Hedlund) The dynamics of transformation groups, Transactions of the American Mathematical Society, vol. 65 (1949), pp. 348-359. 11. The extremum law, Proceedings of the American Mathematical Society, vol. 3 (1952), p. 631.
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1. Sur une propriltk des ensembles minima, Comptes Rendus (Doklady) de l'Academie des Sciences de l'URSS, vol. 14 (1937), pp. 261-262. 2. Sur la theorie des ensembles quasi-minimaux, Comptes Rendus (Doklady) de l'Academie des Sciences de 1'URSS, vol. 15 (1937), pp. 113-116. 3. Sur les mouvements des systbmes dynamiques qui admettent "l'incompressibilite" des
domaines, American Journal of Mathematics, vol. 59 (1937), pp. 803-808. 4. Sur les thEorbmes de recurrence dans la dynamique generate, American Journal of Mathematics, vol. 61 (1939), pp. 149-160. HOPF, E. 1. Zwei Sdtze caber den wahrscheinlichen Verlauf der Bewegungen dynamischer Systeme,
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fiber die Verhandlungen der sAchsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physischen Masse, vol. 91 (1939), pp. 261-304. HoPF, H. 1. Zum Clifford-Kleinschen Raumproblem, Mathematische Annalen, vol. 95 (1926), pp. 313-339. 2. (With W. Rinow) Ueber den Begriff der vollstdndigen diferentialgeometrische Fldche, Commentarii Mathematici Helvetici, vol. 3 (1931), pp. 209-225. HUREWICZ, WITOLD
1. (With Henry Wallman) Dimension Theory, Princeton Mathematical Series No. 4, 1941. KAKUTANI, S.
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TOPOLOGICAL DYNAMICS
146 VON KER.EKJARTO, B.
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1. Riemannsche Mannigfaltigkeiten and nicht euklidische Raumformen, Sitzungsberichte
der Preussischen Akademie der Wissenschaften, (1927), pp. 164-196; (1928), pp. 345-442; (1929), pp. 414-457; (1930), pp. 304-364, 504-541; (1931), pp. 506-534. KURATOWSKI, CASIMIR
1. Topologie I, Warsaw, 1933. KUROSCH, A.
1. Theory of Groups, Moscow, 1944, (Russian). LIE, SOPHUS
1. (With F. Engel) Theorie der Transformationgruppen, Leipzig, 1888. LOBELL, F.
1. Die uberall regularen unbegrenzten Flachen fester Krummung, Dissertation, Tubungen, 1927.
2. Beispiele geschlossener dreidimensionaler Clifford-Kleinscher Resume negativer Krammung, Berichte fiber die Verhandlungen der sachischen Akademie der Wissenschaften
zu Leipzig, Mathematisch-Physischen Klasse, vol. 83 (1931), pp. 167-174. MARK, WILHELM
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1. On central trajectories and a problem of Birkhoff, Matematiceskii Sbornik (N. S.), vol. 26 (68), (1950), pp. 266-290. MARKOFF, A. A.
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INDEX Admissible orbit pairs, 10.38 Admissible subsets of T, 3.12 Almost periodic, 3.38 Asymptotic, 10.21 Asymptotic geodesics, 13.08 Averaging process, 4.75 Axiom
continuity, 1.01 homomorphism, 1.01 identity, 1.01 Axis of a hyperbolic transformation, 13.05 Baire subset, 9.11 Base, 3.29 Bisequence, 12.03 Bisequence space, 12.08 Block, 12.03 Block, n-, 12.04 Bounded, 14.02 Bounded set of mappings, 11.11 Cantor-manifold, 2.16 Center, 7.17 Centralizer of A in G, 9.31 Compactification of a topological group, 4.47 Compactive, 8.11 Composition, 4.74 Contained in, 12.04 Cylinder homeomorphism, 14.03 Decomposition, 1.37 Discrete, 3.15 Discrete in M, 13.41 Dual, 12.27 Dyadic tree, 14.15 Envelope x-, 2.08 Equicontinuous set of mappings, 11.07 Expansive, 10.31 Extension left S-, 1.43 right S-, 1.43 Or, 12.29 Extensive in T, 3.37 Fixed point, 3.06, 10.33 positive, 13.05 negative, 13.05 Flow, 1.30 geodesic, 13.10
geodesic partition, 13.25 horocycle, 13.12 horocycle partition, 13.31 Fundamental region, 13.45 Generative topological group, 6.01 Group motion preserving, 9.28 orbit preserving, 9.30 phase, 1.01 transition, 1.08 Homeomorphism motion preserving, 9.26 orbit preserving, 9.29 periodic, 10.33 pointwise periodic, 10.33 Homeomorphism group, 1.14 discrete, 1.14 topological, 1.12 total, 1.14 Horocycle, 13.07 Hyperbolic area, 13.01 circle, 13.06 distance, 13.05 length, 13.01 line, 13.03 plane, 13.01 ray, 13.03 Indexed, left, 12.46 Indexed, right, 12.46 Intrinsic property, 1.03 Invariant set, 1.22 Isochronous, 3.38 weakly, 3.38 Isometry with axis, 13.05 Isomorphism topological, 1.02 uniform, 1.02 Length (of a block), 12.04 Limit-entire, 13.26 Limit-partial, 13.26 Limit point P-, 6.33 a-, 10.15 co-, 10.15 149
150
INDEX
Limit set, 13.13 P-, 6.33 a-, 10.15 w-, 10.15
Locally bounded set of mappings, 11.30 Mean, 4.81 a-, 4.78
Minimal orbit-closure, 2.11
S-,2.11 totally, 2.27 Mixing, regionally, 9.02 Mobile, 13.22 Motion-projection, 1.08 Motion-space, 1.08 Motion x-, 1.08 Nonseparated, 10.02, 10.38
Normalizer, 9.31 Orbit, 1.26, 10.12 Orbit-closure, 1.26 minimal, 2.11 Paddle motion, 13.05 Partition, 1.27 Period of a homeomorphism, 10.33 of a point, 10.33 of T, 3.04 of T at x, 3.02 Periodic, 3.06 almost, 3.38 at x, 3.06 pointwise, 3.06 regularly almost, 3.38 weakly almost, 3.38 Periodic point, under homeomorphism, 10.33 Permutation of a set, 11.15 Topological group monothetic, 4.49 solenoidal, 4.49 Phase group, 1.01 projection, 1.01 space, 1.01
Point at infinity of a horocycle, 13.07 of an h-ray, 13.03 Points at infinity of an h-line, 13.03 Projection motion, 1.08 phase, 1.01 transition, 1.08
Property intrinsic topological, 1.03 intrinsic uniform, 1.03 Ray left, 12.02 right, 12.02 Recurrent, 3.38, 10.18 Recursive, 3.13 uniformly, 3.42, 3.44, 3.47, 3.50 weakly, 3.42, 3.44, 3.47, 3.50 Regularly almost periodic, 3.38 Replete in T, 3.37 Representation, A-, 12.42 Restriction, 1.32 Reverse, 12.04 Rotor, 13.32 Saturation, 1.34 Semigroup, 2.05 Semiorbit, 10.12 Separated, 10.02, 10.38 Sequence left, 12.03
right, 12.03 Shift transformation, 12.11 Similar, 12.04 Space product, 1.49 Star, (1-, 1.34 Star-closed, 1.35 Star-indexed, 2.35 Star-open, 1.35 Subblock, 12.04 Subgroup restriction, 1.32 Subsequence left, 12.04 right, 12.04 Subspace restriction, 1.32 Symbol class, 12.01 Symbolic flow, 12.11 Syndetic, 2.02, 2.03 Topology
bilateral compact-index, of a set of permutations, 11.34
bilateral compact-open, of a set of permutations, 11.44 bilateral space-index, mutations, 11.16
of a set of per-
compact-index, of a set of permutations, 11.34
inverse compact-index, of a set of permutations, 11.34
inverse compact-open, of a set of permutations, 11.44
INDEX
inverse space-index, of a set of permutations, 11.16 point-index, of a set of mappings, 11.24 space-index, of a set of mappings, 11.01 space-index, of a set of permutations, 11.16 Total power, 4.74 Total homeomorphism group, 1.14 Totally minimal, 2.27 Totally noncompactive, 8.12 Trace, 2.40 Transformation group, 1.01 bilateral, of T, 1.51 discrete, 1.14 effective, 1.09 equicontinuous, 1.52 equicontinuous at x, 1.52 functional, over (X, T, 7r) to Y, 1.68 ,p-inverse partition, 1.42 left, of T, 1.51 left functional, over T to Y, 1.63 left, of T/S induced by T, 1.56 left, of S induced by T under gyp, 1.58 left uniform functional, over T to Y, 1.62 partition, 1.39 (D-orbit partition, 1.40 ''-orbit-closure partition, 1.41 topological, 1.01 right, of T, 1.51 right functional, over T to Y, 1.63 right, of S induced by 1' under p, 1.58 right, of T\S induced by T, 1.56 right uniform functional, over T to Y, 1.62
uniform functional, over (X, T, a) to Y, 1.66
151
uniformly continuous, 4.36 uniformly equicontinuous, 1.52 Transformation subgroup, 1.32 Transient, 13.28 Transition group, 1.08 projection, 1.08 t-, 1.08 Transitive, 9.02 Translate, n-, 12.04 Uniform convergence at a point, 9.34 Uniformity
bilateral compact-index, of a set of permutations, 11.34 bilateral space-index, of a set of permutations, 11.16 compact-index, of a set of mappings, 11.24
compact-index, of a set of permutations, 11.34
inverse compact-index, of a set of permutations, 11.34 inverse
space-index,
of
a
set
of
per-
mutations, 11.16 partition, 2.34 point-index, of a set of mappings, 11.24 space-index, of a set of mappings, 11.01 space-index, of a set of permutations, 11.16 Uniformly equicontinuous set of mappings, 11.08
Union of a sequence of blocks, 12.05 Unitangent, 13.10, 13.48 space, 13.10, 13.48 Universally transitive, 9.02 Weakly recursive, 3.13, 3.42, 3.44, 3.47, 3.50
Colloquium Publications 1.
2.
31. 32.
4.
H. S. White, Linear Systems of Curves on Algebraic Surfaces; F. S. Woods, Forms of Non-Euclidean Space; E. B. Van Vleck, Selected Topics in the Theory of Divergent Series and of Continued Fractions; 1905, xii, 187 pp. $3.00 E. H. Moore, Introduction to a Form of General Analysis; M. Mason, Selected Topics in the Theory of Boundary Value Problems of Differential Equations; E. J. Wilczynski, Projective Differential Geometry; 1910, x, 222 pp. out of print G. A. Bliss, Fundamental Existence Theorems, 1913; reprinted, 1934, out of print ii, 107 pp. E. Kasner, Differential-Geometric Aspects of Dynamics, 1913; reprinted, 1947, ii, 117 pp. 2.50 L. E. Dickson, On Invariants and the Theory of Numbers; W. F. Osgood, Topics in the Theory of Functions of Several Complex Variables; 1914, xii, 230 pp.
out of print
5,. G. C. Evans, Functionals and their Applications. Selected Topics, Including Integral Equations, 1918, xii, 136 pp. out of print 52.
6. 7. 8.
0. Veblen, Analysis Situs, 1922; 2d ed., 1931; reprinted, 1951, x, 194 pp. 3.35 G. C. Evans, The Logarithmic Potential. Discontinuous Dirichlet and Neumann Problems, 1927, viii, 150 pp. out of print E. T. Bell, Algebraic Arithmetic, 1927, iv, 180 pp. out of print L. P. Eisenhart, Non-Riemannian Geometry, 1927; reprinted, 1949, 2.70 viii, 184 pp.
9. 10.
G. D. Birkhoff, Dynamical Systems, 1927; reprinted, 1952, viii, 295 pp. 4.60 A. B. Coble, Algebraic Geometry and Theta Functions, 1929; reprinted, 4.00 1947, viii, 282 pp.
11.
D. Jackson, The Theory of Approximation, 1930; reprinted, 1951,
12. 13.
3.35 out of print R. L. Moore, Foundations of Point Set Theory, 1932, viii, 486 pp. out of print viii, 178 pp. S. Lefschetz, Topology, 1930, x, 410 pp.
14.
J. F. Ritt, Differential Equations from the Algebraic Standpoint, 1932; reprinted, 1947, x, 172 pp. 3.00
15.
M. H. Stone, Linear Transformations in Hilbert Space and their Applications to Analysis, 1932; reprinted, 1951, viii, 622 pp.
8.00
16.
G. A. Bliss, Algebraic Functions, 1933; reprinted, 1947, x, 218 pp.
out of print J. H.-M. Wedderburn, Lectures on Matrices, 1934; reprinted, 1949, x, 205 pp. 3.35 18. M. Morse, The Calculus of Variations in the Large, 1934; reprinted, 1947, x, 368 pp. 5.35 19. R. E. A. C. Paley and N. Wiener, Fourier Transforms in the Complex Domain, 1934; reprinted, 1954, viii, 184 pp. + portrait plate 4.00 20. J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, 1935, x, 382 pp. out of print 21. J. M. Thomas, Differential Systems, 1937, x, 118 pp. out of print 22. C. N. Moore, Summable Series and Convergence Factors, 1938, vi, 105 pp. out of print 23. G. Szego, Orthogonal Polynomials, 1939; reprinted, 1948, x, 403 pp. out of print 24. A. A. Albert, Structure of Algebras, 1939; reprinted, 1952, xii, 210 pp. 4.00 25. G. Birkhoff, Lattice Theory, 1940; enlarged and completely rev. ed., 1948, xiv, 283 pp. 6.00 26. N. Levinson, Gap and Density Theorems, 1940, viii, 246 pp. 4.00 27. S. Lefschetz, Algebraic Topology, 1942; reprinted, 1948, vi, 393 pp. out of print 28. G. T. Whyburn, Analytic Topology, 1942; reprinted, 1948, x, 280 pp. out of print 29. A. Weil, Foundations of Algebraic Geometry, 1946, xx, 288 pp. 5.50 30. T. Rado, Length and Area, 1948, vi, 572 pp. 6.75 31. E. Hille, Functional Analysis and Semi-Groups, 1948, xii, 528 pp. out of print 32. R. L. Wilder, Topology of Manifolds, 1949, x, 402 pp. 7.00 33. J. F. Ritt, Differential Algebra, 1950, viii, 184 pp. 4.40 34. J. L. Walsh, The Location of Critical Points of Analytic and Harmonic Functions, 1950, viii, 384 pp. 6.00 35. A. C. Schaeffer and D. C. Spencer, Coefficient Regions for Schlicht 17.
36.
Functions, with a chapter on The Region of Values of the Derivative of a Schlicht Function by Arthur Grad, 1950, xvi, 311 pp. 6.00 W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, 1955, viii, 151 pp. 5.10 AMERICAN MATHEMATICAL SOCIETY Providence, R. I., 80 Waterman Street
