# An introduction to ergodic theory - Walters P.

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Consider now the north-south map of K. For this map S2(T) = {N,S}. Clearly {N,S} <=. £2{T) since N,S are both fixed points. Let x$ {N,S) and we show x is wandering Choose у between T~ ‘x and x. Then Ту lies between

xand Tx. Let U be the open arc between у and Ту. Then U is a neighbourhood of x and since T~nU is the open arc with end points T~"y, T~n+1y we see that the sets {T~nU}™=0 are pairwise disjoint. Therefore x is wandering and

As mentioned before we show in Theorem 6.15 that if ц is a orobability on the Borel subsets of X and invariant for T then n(Q(T)) = 1. For the north-south map we can use this to find all the invariant probabilities (see §6.4).

One can readily compute C2(T) when T: XA -*■ XA is a topological Markov chain. The calculation is like the division of a Markov chain into equivalent sets of states.

N

Ту

‘Tx

a (T) = {n,s}.

$5.4 Topological Transitivity

127

§5.4 Topological Transitivity

Topological transitivity is a weakening of minimality. Agam X always denotes a compact metric space.

Definition 5.6

(i) A continuous transformation T: X -* X is called one-sided topologically transitive if there exists some x e X with {Т"(х)[и > 0} dense in X.

(ii) A homeomorphism T\X X is called topologically transitive if there is some x e X with 0T(x) = {Tn(x)|n e Z} dense in X.

Both of these concepts make sense for a homeomorphism and we shall see how they are related after giving some equivalent forms of the definitions. Recall that a set which is the intersection of a countable collection of open sets is called a Gt.

Theorem 5.8. The following are equivalent for a homeomorphism T: X —> X of a compact metric space. >

(i) T is topologically transitive.

(ii) Whenever E is a closed subset of X and ТЕ = E then either E = X or E is nowhere dense (or, equivalently, whenever U is an open subset of X with TV = U then U = 0 or V is dense).

(iii) Whenever U, V are non-empty open sets then there exists n e Z with

Tn(U) n УФ0.

(iv) {\ e X:(),(.*) = X} is a dense Сй.

Proof

(i) => (ii). Suppose 07(xo) = X and let E Ф 0, E closed and ТЕ = E. Suppose U is open and U с E,U Ф 0. Then there exists p with Tp(x0) eUczE so that 0r(xo) с: E and X = E. Therefore either E has no interior or E = X.

(ii) => (iii). Suppose U, V Ф 0 are open sets. Then _<*, T"U is an open T-invariant set, so it is necessarily dense by condition (ii). Thus (J“ T"U n V Ф 0.

(iii) => (iv). Let Uu U2, ■ ■ ■, U„,... be a countable base for X. Then {хбХ|ад = X} = n„”= 1 Um=-co TmU„ and Um=-=o T\UJ is Clearly dense by condition (iii). Hence the result follows.

(iv) => (i). This is clear. □

For the analogous theorem for one-sided transitivity we assume TX = X. Note that if E с X the condition E c T~lE is equivalent to ТЕ c £.

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5 Topological Dynamics

Theorem 5.9. The following are equivalent for a continuous transformation T:X-*X with TX = X. f

(i) T is one-sided topologically transitive.

(ii) Whenever E is a ct&sed subset of X and E a T_1E then either E — X or E is nowhere dense (equivalently, whenever U is an open subset of X and T~lU a U then U = 0 or U is dense).

(iii) Whenever U, V are non-empty open sets there exists n ;> 1 with T~nU n V Ф 0.

(iv) The set of points x with {T"(x)|n > 0} dense in X is a dense G6. Proof

(i) =» (ii). Suppose {T"(x0) | n > 0} is dense in X, and suppose E is closed and ТЕ с E. Suppose U m a non-empty open set with U с E. Then Tp(x0) e U for some p ^ 0, so that {Тп(х0)|и > p} a E. Therefore

{x0, T(x0),.. ., Tp~ ‘(x0)} u £ = X.

By applying T to each side we get {T(x0),..., Tp-1(x0)} u E = X so by repeated application of T we have E = X. So if E has interior then E = X.

(ii) => (iii). Suppose U, V are non-empty open sets. Then (J“=1 T~nU is open and T-HUn^i T~nU) c (J»., T~nU so (Jn°°=1 T~HU is dense by (ii). Therefore T~nU n V Ф 0 for some n > 1.

(iii) => (iv). If is a base for the topology then {x| {T"(x)}“=0 is dense} = f)*=1 Q“=0 T~mUn. By (iii) we know (J“=0 T~mU„ is dense so the result follows.

(iv) =» (i). This is clear. □

The assumption 1 X = X was made because of the type of behaviour occurring in the following example. Let X = {\/n\n ;> 1} u {0} with the induced topology from the real line. Define T.X -»X by T{0) = 0 and T(l/n) = l/(n + 1). Hence T moves each point 1 /n to the'next point on the left. Only the point 1 has a dense forward orbit so statements (i) and (iv) are not equivalent for this example. Also if E = X\{1} then E a T~1E and E is closed so (ii) is violated.

We now consider the connection between the two types of transitivity when T is a homeomorphism.

Theorem 5.10. Let T.X^X be a homeomorphism. Then T is one-sided topologically transitive iff T is topologically transitive and Q(T) = X.

Proof. Suppose {T"(x0)|n > 0} is dense in X. Clearly T is topologically transitive. If Q(T) ф X there is a non-empty open set U such that {T"U\neZ} are pairwise disjoint sets. For some n0 > 0, T"°(x0)g U. Therefore Tn+'l°(xn)e T"U,n> 0, so that only {x0, T(x0),..., Tn°~ ^Xq)} can belong to (J,“ ,7''U

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