Shrinking Targets for Countable Markov Maps

Abstract

Let $T$ be an expanding Markov map with a countable number of inverse branches and a repeller $Λ$ contained within the unit interval. Given $α\in \R_+$ we consider the set of points $x \in Λ$ for which $T^n(x)$ 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 $-s\log|T'|$ 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 $\barΛ=[0,1]$ the same holds true for all $y \in [0,1]$. However, given $β\in (0,1)$ we provide an example of an expanding Markov map $T$ with a repeller $Λ$ of Hausdorff dimension $β$ with a point $y\in \barΛ$ such that for all $α\in \R_+$ the dimension of the shrinking target set is zero.