Teaching Assistant Work (Ergodic Theory)

1 First Lecture on Entropy Example 1.1 Consider the unilateral shifts f : {1, 2}N → {1, 2}N and g : {1, 2, 3}N → {1, 2, 3}N equipped with the Bernoulli measures μ and ν, respectively. Show that, for any set X ⊂ {1, 2}N satisfying f−1(X) = X and μ(X) = 1, there exists x ∈ X such that #(X ∩ f−1(x)) = 2. Conclude that if k 6= l then (f, μ) and (g, ν) can not be ergodically equivalent. Proof. Denote Xi = X ∩ [0; i] and pi = μ([0; i]), for i = 1, 2. As μ is a Bernoulli measure, f(X1) = f(X2) = f(X) has full μ measure. Now take x ∈ f(X). So, x has two pre-images. Similarly, for the system (g, ν), we can find a set g(Y1) ∩ g(Y2) ∩ g(Y3) with full measure, such that any point y in it has three pre-images. From here, one can easily show that it is impossible for these two systems to be ergodically equivalent.