What question did this study set out to answer?

The aim is to explore recurrence properties in dynamical systems with exponentially decaying correlations.

March 4, 2026

Quantitative recurrence properties and strong dynamical Borel–Cantelli lemmas for dynamical systems with exponential decay of correlations

Key Points

The aim is to explore recurrence properties in dynamical systems with exponentially decaying correlations.
Investigated measure-preserving dynamical systems on the interval [0,1]^d
Analyzed the decay of correlations for Hölder continuous functions
Developed a strong dynamical Borel–Cantelli lemma for recurrence using sequences of hyperrectangles
Established a recurrence lemma applicable to various dynamical systems
Showed that the density function h is integrable with respect to the d-dimensional Lebesgue measure
Validated the findings for specific systems including Gauss maps and toral endomorphisms

Abstract

Abstract Let (0, 1ᵈ, T, ) be a measure-preserving dynamical system so that the correlations decay exponentially for Hölder continuous functions. Suppose that is absolutely continuous with a density function h Lq (Lᵈ) for some q>1, where Lᵈ is the d -dimensional Lebesgue measure. Under suitable conditions on the underlying dynamical system, we obtain a strong dynamical Borel–Cantelli lemma for recurrence: for any sequence \Rₙ\ of hyperrectangles centered at the origin, with sides parallel to the axes and diameter going to 0 as n, where x 0, 1ᵈ and Rₙ+ x is the translation of Rₙ. The result applies to the Gauss map, -transformations, and expanding toral endomorphisms.

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