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Dynamical Borel–Cantelli lemma for recurrence theory

journal contribution
posted on 21.06.2021, 06:54 by Mumtaz Hussain, Bing Li, David Simmons, Baowei Wang

We study the dynamical Borel–Cantelli lemma for recurrence sets in a measure-preserving dynamical system (X, μ, T ) with a compatible metric d. We prove that under some regularity conditions, the μ-measure of the following set R(ψ) = {x ∈ X : d(T nx, x) < ψ(n) for infinitely many n ∈ N} obeys a zero–full law according to the convergence or divergence of a certain series, where ψ : N → R+. The applications of our main theorem include the Gauss map, β-transformation and homogeneous self-similar sets.

History

Publication Date

12/04/2021

Journal

Ergodic Theory and Dynamical Systems

Pagination

(p. 1-15)

Publisher

Cambridge University Press (CUP)

ISSN

0143-3857

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