Quantitative recurrence properties of expanding maps

2007/03/08 by José Fernández, Fernandez, J. L., María V. Melián +3 · 2 citations
Mathematics · #11K55 #11K60 #28D05 #30D05 #30D50 #37A25 #37D05 #37F10 #Advanced Differential Equations and Dynamical Systems #Advanced Topology and Set Theory #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.math/0703222

openalex publication_date 2007/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Under a map T, a point x recurs at rate given by a sequence r_n near a point x₀ if d(Tⁿ(x),x₀)< r_n infinitely often. Let us fix x₀, and consider the set of those x's. In this paper, we study the size of this set for expanding maps and obtain its measure and sharp lower bounds on its dimension involving the entropy of T, the local dimension near x₀ and the upper limit of 1/n log 1/r_n. We apply our results in several concrete examples including subshifts of finite type, Gauss transformation and inner functions.

Citations

• Recurrence times and rates of mixing

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• The shrinking target and recurrence problem for non-autonomous systems
• Quantitative recurrence properties in conformal iterated function systems

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