Ergodic Theory: Recurrence

Almost every, essentially: Given a Lebesgue measure space (X,B, μ), a property P (x) predicated of elements of X is said to hold for almost every x ∈ X, if the set X \ {x : P (x) holds} has zero measure. Two sets A,B ∈ B are essentially disjoint if μ(A ∩B) = 0. Conservative system: Is an infinite measure preserving system such that for no set A ∈ B with positive measure are A,T−1A,T−2A, . . . pairwise essentially disjoint. (cn)-conservative system: If (cn)n∈N is a decreasing sequence of positive real numbers, a conservative ergodic measure preserving transformation T is (cn)-conservative if for some non-negative function f ∈ L1(μ), ∑∞ n=1 cnf(T nx) = ∞ a.e. Doubling map: If T is the interval [0, 1] with its endpoints identified and addition performed modulo 1, the (non-invertible) transformation T : T → T, defined by Tx = 2x mod 1, preserves Lebesgue measure, hence induces a measure preserving system on T. Ergodic system: Is a measure preserving system (X,B, μ, T ) (finite or infinite) such that every A ∈ B that is T -invariant (i.e. T−1A = A) satisfies either μ(A) = 0 or μ(X \ A) = 0. (One can check that the rotation Rα is ergodic if and only if α is irrational, and that the doubling map is ergodic.)