2011/07/24 by Henry W. J. Reeve, Henry WJ Reeve, Reeve, Henry WJ · 1 citation
Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #math.DS
paper · pdf · doi:10.48550/arxiv.1107.4736
25 pages
openalex publication_date 2011/07/24 · arxiv created 2011/09/12 · arxiv updated 2011/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let T be an expanding Markov map with a countable number of inverse branches and a repeller Λ contained within the unit interval. Given α∈ \R+ we consider the set of points x ∈ Λ for which Tn(x) hits a shrinking ball of radius e-nα around y for infinitely many iterates n. Let s(α) denote the infimal value of s for which the pressure of the potential -slog|T'| is below s α. Building on previous work of Hill, Velani and Urbański we show that for all points y contained within the limit set of the associated iterated function system the Hausdorff dimension of the shrinking target set is given by s(α). Moreover, when Λ=[0,1] the same holds true for all y ∈ [0,1]. However, given β∈ (0,1) we provide an example of an expanding Markov map T with a repeller Λ of Hausdorff dimension β with a point y∈ Λ such that for all α∈ \R+ the dimension of the shrinking target set is zero.