# An introduction to ergodic theory - Walters P.

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Example 1. Consider the 2-torus K2 and identity (z, w) e K2 with (I, w). i.e. an element of K2 is identified with its group inverse. This identification is two-to-one except at the four points (1,1), (1,-1), (—1,1) ana (-1,-1) which are their own group inverses. The identification space with the quotient topology is homeomorphic to the 2-sphere S2. Let ф:К2 -*S2 be the projection. Let A:K2-*K2 be a continuous automorphism. Since A maps equivalence classes to equivalence classes it induces a homeomorphism T:S2 -* S2. Clearly 7 is a factor of A. We shall see later that A is expansive if the matrix [/1] has no eigenvalues of unit modulus. However the homeomorphism T:S2 -* S2, induced by such an expansive A: К2 -* K2, is not expansive. To see this let us use additive notation on K2, so the identification means (x, y) + Z2 is identified with (—x, — y) + Z2. Let Vs, Vu be the eigenspaces in R2 corresponding to the eigenvalues As, /„ of the linear transformation A where |as| < 1 and |A„| > 1. (s denotes stable and и denotes unstable.) Let e > 0 be given and choose any point (x, y) in the Euclidean ball in R2 with centre (0,0) and radius e. Consider the parallelogram deter-

mined by the translates of Vs, Vu that go through (x, y) and those that pass through (—x, — y). The other vertices have the form (u, v) and ( — u, — v) and all the vertices are contained in the Euclidean ball of radius ce for

§5.6 Expansive Homeomorphisms

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some constant с depending only on the slopes of Vs and Vu. Nonce that | Л"(х, J’) ~ A"(u, d)|| = i£||(x, у) - (и, i)\\ if n < 0 since (u, v) - (x, y) e Vu, and | Л"(х, y) - A”( — u, —1;)|| = K\\(x, y) - (-u, -d)|| if n > 0 since (-u, -d) -(к, y) e Vs. So if d is the metric on the torus induced from ||-||, and if we write (x, y) rather than (x, y) + Z2 as a point of K2, we have d(An(x, y), An(u, v) j < 2ce V/i £ 0 and rf(4n(x,y), A"( — u, —v)) < 2ce Vn 2 0. If T:S2 -* S2 had a generator у — {С,,..., Ck} then ф~1у would have the property that each set Пп°= -» А~пф~1С1п contains at most one equivalence class. Choose г so that 2ce is a Lebesgue number for the open cover ф~ ly of K2. The above shows that some set „ А~”ф~ lCin contains the equivalence class of (x, y) and the equivalence class of (и, v), contradicting the fact that it contains at most one equivalence class.

Example 2. Let Tz = az be a minimal rotation of K. We shall represent T as a factor of a subset of the two-sideo shift on two symbols. Consider the cover of К by the closed intervals (arcs) between — 1 and 1 on K. Call one of them A0 and the other A1. If z e K\{an, —an:,neZ} we can uniquely asso-

ciate a member of П-ос {0,1} to z by z -+ {a,,}-* if T”z e Aan. Let Л denote the subset of П-*> {0,1} arising in this way. Let \p:K\{an, —a":neZ}-*A be the map defined above. We want to show ф is injective and the inverse map can be extended to a continuous map ф:Л~* K. We do this by proving for each e > 0 there is an integer N such that if x, у e K\{an, —an:neZ} and (^(x))„ = (ф(у))„ for |n| < N then d(x, y) < e. Suppose e > 0 is given. Choose N > 0 so that {1,а±1,й±2,...,я±л'} is e/2-dense in K. Suppose x, з e —a":n e Z} and (^(x))„ = (ф(у))п for |и| < N. We shall show

d(x, y) < e. The assumption (ф(х))„ —(ф(у))„, |и| < N means that anx and a"y belong to the same element of the cover for |n| < N. If у = — x then this clearly cannot happen. So suppose the counter-clockwise distance from у to x is smaller than the clockwise distance. For some n with (n| < N a"x is in the open interval of hngth e starting at 1 and going counter-clockwise. Hence iТу must also be in the upper half of the circle and by the assumption about the relative positions of x and y, any must be between 1 and a"x. Hence d(a”x,a!‘y) < e and so d(x, y) < e.

Since фТ = Sip, where S denotes the shift, we have фБ(х) = Тф(х) Vx e A where ф:Л~* К denotes the extension of ф~1. The continuous map ф is

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5 Topulogical Dynamics

surjective because ф(Л) is a closed subset of К containing the dense set

X1. We shall see later that £ is an expansive homeonormphism whereas T, being an isometry, is not expansive.

We shall now show that every expansive homeomorphism is a factor of a sub^t of a two-sided shift.

We shall need to use the following result which is similar to half of Theorem 5.21. Recall that the diameter of a cover is the supremum of the diameters of its members.

Theorem 5.23. Let T be an expansive homeomorphism of a compact metric space (X,d) and let 8 be an expansive constant. Let у be a finite cover of X (not necessarily an open cover) by sets {C,,.. ., Cr} with Qiam (C;) < <5, 1 < j <r. Then diam(\/"= T~jy) -* 0 as n -+ oo.

Proof. Suppose the conclusion is false. There exists e0 > 0 a subsequence {и,} of natural numbers, points x;, y, with d(xit y;) > e0 and xf, yt e P)"'= T~JC,} where Cimj e y. We can choose a subsequence {ik} of natural members such that xik —► x and yik -* y. Hence d(x, y) > £0. Consider the sets Cik 0. Infinitely many of them coincide so for some C,0 e y, xik, yik e C,0 for infinitely many k. Therefore x, у e C,0. Similarly, for each j infinitely many of the sets Cf coincide so there is some Ctj e у with x, _y e T~]Ct.. Therefore d(TJx, Tjy) < 8 V/ e Z ana so x = y, contradicting d(x, y) > e0. □

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