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By K. Mathiak, Steffen Supra

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Consider the 2-to-1 map T : R → R defined by T (x) = 1 2 x− 1 x for x = 0, and T (0) = 0. For any L1 function f , the substitution y = T (x) shows that ∗ The topology on X is simply the product topology, which is also the metric topology given by the metric defined by d(x, y) = 2−k where k = max{j | xi = yi for |j| k} if x = y and d(x, x) = 0. In this metric, points are close together if they agree on a large block of indices around 0 ∈ Z. 1 Measure-Preserving Transformations ∞ f (T (x)) −∞ dx = π(1 + x2 ) 19 ∞ f (y) −∞ dy π(1 + y 2 ) (in this calculation, note that T is only injective when restricted to (0, ∞) or (−∞, 0)).

Thus μ(B) = 0, so there is a set F1 ⊆ E with μ(F1 ) = μ(E) and for which every point of F1 returns to E at least once under iterates of T . The same argument applied to the transformations T 2 , T 3 and so on defines subsets F2 , F3 , . . of E with μ(Fn ) = μ(E) and with every point of Fn returning to E under T n for n 1. The set Fn ⊆ E F = n 1 has μ(F ) = μ(E), and every point of F returns to E infinitely often. Poincar´e recurrence is entirely a consequence of the measure space being of finite measure, as shown in the next example.

3) =⇒ (4): Let A and B be sets of positive measure. By (3), ∞ μ T −n A = 1, n=1 so ∞ 0 < μ(B) = μ B ∩ T −n A n=1 It follows that there must be some n ∞ μ B ∩ T −n A . n=1 1 with μ(B ∩ T −n A) > 0. 3 Ergodicity 25 (4) =⇒ (1): Let A be a set with T −1 A = A. Then 0 = μ(A ∩ X A) = μ(T −n A ∩ X A) for all n 1 so, by (4), either μ(A) = 0 or μ(X A) = 0. (2) =⇒ (5): We have seen that if (2) holds, then T is ergodic. Let f be a measurable complex-valued function on X, invariant under T in the stated sense.

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Algebra[Lecture notes] by K. Mathiak, Steffen Supra

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