Ergodic Theory: Recurrence

Abstract

write µ = µ t dλ(t), where λ is a probability measure on [0, 1] and µ t are Tinvariant probability measures on (X, X ) such that the systems (X, X , µ t , T ) are ergodic for t ∈ [0, 1].Ergodic theorem: States that if (X, B, µ, T ) is a measure preserving system and f ∈ L 2 (µ), then lim N →∞= 0, where P f denotes the orthogonal projection of the function f onto the subspace {f ∈ L 2 (µ) :Hausdorff a-measure: Let (X, B, µ, T ) be a measure preserving system endowed with a µ-compatible metric d.The Hausdorff a-measure H a (X) of X is an outer measure defined for all subsets of X as follows: First, for, where the infimum is taken over all countable coverings of A by sets U i ⊂ X with diameter r i < ε.Then define H a (A) = lim sup ε→0 H a,ε (A).Infinite measure preserving system: Same as measure preserving system, but µ(X) = ∞.Invertible system: Is a measure preserving system (X, B, µ, T ) (finite or infinite), with the property that there exists X 0 ∈ X, with µ(X\X 0 ) = 0, and such that the transformation T : X 0 → X 0 is bijective, with T -1 measurable.Measure preserving system: Is a quadruple (X, B, µ, T ), where X is a set, B is a σ-algebra of subsets of X (i.e.B is closed under countable unions and complementation), µ is a probability measure (i.e. a countably additive function from B to [0, 1] with µ(X) = 1), and T : X → X is measurable (i.e.T -1 A = {x ∈ X : T x ∈ A} ∈ B for A ∈ B), and µ-preserving (i.e.µ(T -1 A) = µ(A)).Moreover, throughout the discussion we assume that the measure space (X, B, µ) is Lebesgue (see Section 1.0 of [3]).µ-compatible metric: Is a separable metric on X, where (X, B, µ) is a probability space, having the property that open sets measurable.Positive definite sequence: Is a complex-valued sequence (a n ) n∈Z such that for any n 1 , . . ., n k ∈ Z and z 1 , . . ., z k ∈ C, k i,j=1 a n i -n j z i z j ≥ 0. Rotations on T: If T is the interval [0, 1] with its endpoints identified and addition performed modulo 1, then for every α ∈ R the transformation R α : T → T, defined by R α x = x + α, preserves Lebesgue measure on T and hence induces a measure preserving system on T. Syndetic set:, where Λ ⊂ Z (assuming the limit to exist).Alternatively for measurable E ⊂ R m , D(E) = lim sup l(S)→∞ m(S∩E) m(S) , where S ranges over all cubes in R m , and l(S) denotes the length of the shortest edge of S.

Locations

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