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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p.
Download (direct link): anintroduction1982.djvu
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Theorem 5.20. If T:X -+ X isa homeomorphism of a compacnnetrisable space then T has a generator iff T has a weak generator.
S Topological Dynamics
Proof. A generator is clearly a weak generator. Now suppose ft is a weak generator for T, (} = {B^ ... ,BS}, and let S be a Lebesgue number for (} (see Theorem 0.20). Let a be a finite open cover by sets А/ having diam(y4,) < S. So if Ain is a bisequence in a then Vh 3l/„ with Ain £ Bjn. Hence
П-» ГМ,сП^Г"В,,
which is either empty or a single point. So a is a generator. □
The following shows that a generator determines the topology on X. If a, P are open covers of X then a v /? is the open cover of X by the sets A n B, A e a, В g B. T~la is the open cover by the sets T~1 A, A fc a.
Theorem 5.21. Lei T:X -» X be a homeomorphism of a compact metric space (X,d). Let a be a generator for T. Then Vf > 0 3N > 0 such that each set in V-n T~"a has diameter less than e. Conversely, VN > 0 Зг > 0 such that d(x, y) < e implies
x, у G П T~”An
for some A _w,..., AN G a.
Proof. Suppose the first part of the theorem does not hold. Then Зг > 0 such that V/ > 0 Зх7, yjt d(xj, ys) > г and 3 AJtl g a, —j<i<j with Xj, ij g f]{=-j T~‘Ajti. There is a subsequence {jk} natural numbers such that xjk —v x and yJk —v у since X is compact. We have хФ y. Consider the sets Aj„.o- Infinitely many of them coincide since a is finite. Thus xJk, yJk G A0, say, lor infinitely many к and hencc x, у e Av. Similarly, for each rt, infinitely many AJki„ coincidc and wc obtain A„ e a with .v, у 6 7 A„. Thus
x, у g П Т~ЛАп,
— CO
contradicting the fact that a is a generator.
To prove the converse let N > 0 be given. Let S > 0 be a Lebesgue number for a. Choose e > 0 such that d(x, y) < e implies d(T‘x, T‘y) <6 for — N <, i < N. Hence if d(x, y) < e and |i| < N then Tlx, T‘y e At for some A{ g a. Hence
x’ У е П T~lAi. □
Generators are connected with expansive homeomorphisms which were studied for several years before generators were introduced.
Definition 5.11. A homeomorphism T of a compact metric space (X,d) is said to be expansive if 3<5 > 0 with the property that if x # у then 3n g Z with d(T”x, T”y) > S. We call <5 an expansive constant for T.
55.6 Expansive Homeomorphisms
Remark. Another way to give this definition is аз follows. Consider X x X with T x T acting on it. Define a metric D on X x X by D((u,v), (x,y)) = max{rf(u, x), d(v, y)}. Then T is expansive iff 35 > 0 such that if (x, y) is not an element of the diagonal, then some power of T x T takes (x, y) out of the d-neighbourhood of the diagonal.
The following thcoiem is due to Reddy [1] and Keynes and Robertson [1]. 72
Theorem 5. . Let T be a homeomorphism of a compact metric space (X,d). Then T is expansive iff T has a generator iff T has a weak generator.
Proof. By Theorem 5.20 it suffices to show Tis expansive iff T has a generator.
Let 6 be an expansive constant for T and a a finite cover by open balls of radius 5/2. Suppose x, у e П- со T~lA„ where A„ e a. Then d(Tnx, T”y) <; S Vn e Z so, by assumption x = y. Therefore a is a generator.
Conversely, suppose a is a generator. Let 5 be a Lebesgue number for a. If d(Tnx, T"y) < S V« then V« 3An e a with T"x, T"y e A„ and so,
x> У 6 П T~"A„-
— CO
Since this intersection contains at most one point we have x T is expansive.
Corollary 5. .1
(i) Expansiveness is independent of the metric as long as the metric gives the topoloiiv of X. (However the expansive constant does change.)
(ii) // к ф 0 tlwn T is expansive iff Tk is expansive
(iii) Expansiveness is a topological conjugacy invariant i.e. if, for i = 1,2, Т,-: A’, —► Xt is a homeomorphism of a compact metrisable space and if (j>:Xj -► X2 is a homeomorphism with фТ1 = Т2ф then Tj is expansive iff T2 is expansive.
(i) This is because the concept of generator does not depend on the metric.
(ii) If a is a generator for T then
av T’_1av - • - v T^{k~l)a
is a generator for Tk. Also any generator for Tk is a generator for T.
(iii) A cover a is a generator for T2 iff <^_1a is a generator for Ti. □
Remarks. We make some more comments on how expansiveness behaves relative to natural ways of getting a new homeomorphism from an old one.
(i) If T:X -» X is expansive and У is a closed subset of X with TY = Y then T\r is expansive (i.e. a subsystem of an expansive system is expansive).
=- y. Hence □
5 Topological Dynamics
(2) If Tf -.Xi -» X(, i = 1,2, are expansive then so is 7, x T1:Xl x X2 -*
X, x X2. This extends to finite products but not to infinite products.
(3) If 7i:X/-> Xf, /'=1,2 are homeomorphisms and if </>:X, -► X2 is a continuous map of X, onto X, with $7, = Т2ф then 7, is said to be a factor of Tl. It is clear that if 7, is minimal or topologically transitive or has a dense set of periodic points then any factor T2 also has the corresponding property. However expansiveness is not preserved under the operation of taking factors as the following examples show.
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