Lecture 12 Ergodicity and Mixing February 11, 2008 1 Ergodicity Ergodicity studies the statistical properties of a dynamical system. The idea is that a measure is ergodic if asymptotically the time average of any oberservable (function) is the space average. (i.e. for A a measurabel set Pn i µ(A) = limn!1( i=1 χA(f (x)))=n) for almost every x 2 A.) Let T : X ! X be a measurable transformation and f : X ! R (could be C in general). Then we define (T∗f)(x) = f(T (x)) so T∗f is a function from X to R. We want to look at the statistical independence from f to n T∗ f. Let T be a measure preserving transformation on (X; B; µ) to itself. A function f : X ! R is essentially T -invariant if n µfx 2 X j f (T (x)) 6= f(x)g = 0 for all n 2 N: 1 An example of an essentially T -invariant function is T = Em, X = S with Lebesgue measure, and f(x) = 0 for all x 2 S1. A set A is A essentially T -invariant if χA is essentially T -invariant. So µ(T −1(A) 4 A) = 0 where A 4 B = (A − B) [ (B − A). Definition 1.1 A measure preserving transformation T is ergodic if any es- sentially T -invariant set has measure 0 or full measure. An equivalent definition is T is ergodic if any essentially T -invariant mea- surable function is constant mod 0.(Exercise) 1 Proposition 1.2 Let (X; B; µ) be a measure space and f : X ! R essen- tially invariant for a measure preserving transformation T on X. Then there exists an invariant measurable function f~ such that f~ = f mod 0. n Proof. Let An = fx 2 X j f(T x) = f(x)g. So µ(An) has full measure for all T1 n 2 N. Let Af = n=1 An. Then Af has full measure and Af is T -invariant. Let f(y) if T nx = y 2 A for some n f~(x) = f 0 else If T nx = y and T sx = z, then y and z are in the same orbit, so f~(y) = f~(z). Hence, f~ is well-defined. 2 2 Mixing Definition 2.1 A measure preserving transformation T is mixing on a prob- ability space if lim µ(T −n(A) \ B) = µ(A)µ(B) for any A; B 2 B: n!1 An equivalent definition is T is mixing if Z Z Z lim f(T n(x))g(x)dµ = f(x)dµ g(x)dµ n!1 X X X for any bounded measurable functions f and g. Definition 2.2 A measure preserving transformation T is weak-mixing on a probability space if n−1 1 X lim jµ(T −i(A) \ B) − µ(A)µ(B)j = 0: n!1 n i=0 Equivalently, for any bounded measurable functions f and g we have n−1 1 X Z Z Z lim j f(T i(x))g(x)dµ − fdµ gdµj = 0: n!1 n i=0 X X X Proposition 2.3 Mixing implies weak-mixing which implies ergodicity. 2 Proof. Let T be a measure preserving transformation. If jµ(T −i(A) \ B) − µ(A)µ(B)j ! 0, then so does the average. Proving that mixing implies weak-mixing. Let A be an invariant measurable set. Then for weak-mixing with A = B we have µ(A) = µ(A)2 so µ(A) = 0 or 1. 2 Now we show a connection between measure theoretic assumptions of mixing if the function f is continuous. Proposition 2.4 Let X be a compact metric space and T : X ! X be a continuous map and µ a T -invariant Borel measure on X. 1. If T is ergodic, then the orbit of µ-almost every point is dense in supp(µ). 2. If T is mixing, then T is topologically mixing on supp(µ). Proof. Let T be ergodic. Let U be a non-empty open and invariant set so µ(A) = 1. Then the forward orbit of almost every point visits U. So the set of points whose forward orbit visits every element of a countable basis has full measure in X. Let U and V be open such that µ(U); µ(V ) > 0. Since T is mixing we know that lim µ(T −n(U) \ V ) = µ(U)µ(V ) > 0: n!1 So there exists an n 2 N such that for any n ≥ N we have µ(T −n(U)\V ) > 0 and hence T −n(U) \ V 6= ;. 2 3.