# An introduction to ergodic theory - Walters P.

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§5.4 Topological Transitivity

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contradict.ng the fact that each set T 'U must conta.n an element of the dense set {T"(x0)|h > 0}. Therefore Q(T) = X.

Now suppose T is topologically transitive and Q(T) = X. We use (iii) of Theorem 5.9 to show T is one-sided topologically transitive. Let U, V be non-empty open sets. We want to find some к > 1 with T~kU n V Ф 0. By

(iii) of Theorem 5.8 we know there is some N e Z with W = TNU n V Ф 0 so we may as well suppose N ^ 0. Since Q(T) — X Theorem 5.7 gives the existence of n > N + 1 with T~nW n W Ф 0. Then T~("~N)U n V => T~nW n W Ф 0 so we can take к = n — N. □

Remarks

(!) An example of a topologically transitive homeomorphism which is not one-sided topologically transitive is the following. Let X = {0} и {1} и {(1 /н)|#i > 2} и {1 — (l/n)|n > 2} with the induced topology as a subset of R. Define T:X -» X by T(0) = 0, T(l) = 1 and T maps any other point to the next point on the left. Then if x $ {0,1} the set {7’n(x)|n e Z) is dense, so that 7 is topologically transitive. Clearly T is not one-sided topologically transitive. Notice in this case that £2(7’) = {0} u {1}.

(2) In an analogous way we could define a homeomorphism T:X -» X to be one-sided minimal if { T"(x) | n > 0} is dense in X for each x e X. Then one can show that T is one-sided minimal iff T is minimal. The “only if” part of this is trivial and the “if” part follows fiom the following. One-sided minimality is equivalent to X = (J*=0 T~nU for each non-empty open set U. If U is open and non-empty then the minimality implies T~kU = X. Since X is compact wc have X = Tk'U u TklU u ■ • • u TkrU for some integers kj. Choose N > 0 so that N > ]/с7-|, 1 < j <, r and thon X = T~NX = T " '*k,U u • • • u T~N+krU so X = Ц%0 T~nU. Hence minimality implies one-sided minimality. Of course Q(T) = X when T is minimal, because it contradicts compactness of X to have an open set U with pairwise

disjoint and ^aiTnU = X. □

Topologically transitive homeomorphisms enjoy some of the properties of minimal homeomorphisms and also allow other interesting things to occur; e.g., a dense set of periodic points. We know from Theorem 1.11 that an ergodic affine transformation of a compact, connected, metric abelian group is topologically transitive (even one-sided topologically transitive). We now see that some of them have a dense set of periodic points.

Theorem 5.11. Let A:Kn -» Kn be an ergodic automorphism of the n-torus K". The periodic points of A are exactly those points (»vb ... ,w„)e Kn where each w, is a root of unity. (In adaitive notation these are the points of Rn/Zn of the form (xlt..., x„) + Z" where each xf is rational.) Even if A is not ergodic these points are periodic for A so every automorphism A has a dense set of periodic points.

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5 Topological Dynamics

Proof. Let A be an automorphism Let w = (w!.....wJeX" be so that

each w; is a root of unity. There is some к > 1 with wk = e, the identity element. For each fixed к the set У* = {z g /C":z* = e} is a finite subgroup of K" and AYk = Yk. Hence each member of Yk is a periodic point and so our original w is

Now suppose A is ergodic. We shall use additive notation for this part of the proof, If x + Z" g KnIZn is fixed by Ak then Akx = x + p for some p g Z”. In matrix notation this equation becomes

Since A is ergodic the matrix [/]* — / is an invertible matrix of integers and so its inverse has rational entries. Therefore each x-t is rational. Therefore each periodic point is of the form x + Z" where the coordinates of x are rational. □

Theorem 5.12. The two-sided and one-sided shifts have a densp set of periodic points. For the two-sided shift {x,,}^ is fixed by Tp iff x„ = x„+p Vn G Z. For the onesided shift {x„}o is fixea by Tp iff x„ = xn+p V« > 0.

Proof. We shall consider only the two-sided case. Ifx = {x,}^ and Tpx = x then xp+i = X; for each In other words, the points fixed by Tp have the form (... .Xp-jXoX,,... .Xp-jXoXj,... .Xp-jXoX!,... .Xp-jXo,...) where we have free choice of x0, x,,..., xp_ Therefore the periodic points are dense. □

Parts (ii) and (iii) of Theorem 5.8 show that topological transitivity is (in some sense) a topological analogue of ergodicity. Aleo, topologically transitive homcomorphisms are “indecomposable;” i.e., we cannot write

X = (J Ex, TEX — Ea and Ea closed

a

when T is topologically transitive. So it seems that topologically transitive homeomorphisms are better building blocks than minimal homeomorphisms. If T has a decomposition into minimal pieces then each piece is also topologically transitive. Not all homeomorphisms can be decomposed into topologically transitive pieces, see the example in Remark (1) above. Two important cases where a decomposition is possible are the following. A distal homeomorphism T: X -» X (i.e. for every pair x ф у there exists S = <5(x, y) > 0 with d(T"(x), T"(y))> 8 Vn g Z) can be decomposed into minimal pieces i.e. X = (Jie/ Xt where I is some index set, the sets Xt are pairwise disjoint, closed, TX{ = Xt and T\Xi is minimal and distal (Ellis [1]). If T:M -*M is an Axiom A diffeomorphism of a compact manifold M then Лот is very important. It turns out that Q(T) = (J;=j Q-, where the are

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