Shrinking Targets for Countable Markov Maps

Abstract

Let $T$ be an expanding Markov map with a countable number of inverse branches and a repeller $\Lambda$ contained within the unit interval. Given $\alpha \in {\mathbb{R}}_{+}$ we consider the set of points $x\in \Lambda$ for which $T^n(x)$ hits a shrinking ball of radius $e^{-n\alpha }$ around $y$ for infinitely many iterates $n$. Let $s(\alpha )$ denote the infimal value of $s$ for which the pressure of the potential $-s\log |T^{\prime }|$ is below $s\alpha$. 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 $s(\alpha )$. Moreover, when $\bar{\Lambda }=[0,1]$ the same holds true for all $y\in [0,1]$. However, given $\beta \in (0,1)$ we provide an example of an expanding Markov map $T$ with a repeller $\Lambda$ of Hausdorff dimension $\beta$ with a point $y\in \bar{\Lambda }$ such that for all $\alpha \in {\mathbb{R}}_{+}$ the dimension of the shrinking target set is zero.

25 pages

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