Shrinking Targets for Countable Markov Maps

Let $T$ be an expanding Markov map with a countable number of inverse branches and a repeller $\mathrm{\Lambda }$ contained within the unit interval. Given $\alpha \in {\mathbb{R}}_{+}$ we consider the set of points $x\in \mathrm{\Lambda }$ 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 $-s\log \mathrm{\mid }{T}^{\mathrm{\prime }}\mathrm{\mid }$ is below $s\alpha$. Building on previous work of Hill, Velani and Urba'{n}ski 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 $\bar{\mathrm{\Lambda }}=[0,1]$ 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 $\mathrm{\Lambda }$ of Hausdorff dimension $\beta$ with a point $y\in \bar{\mathrm{\Lambda }}$ such that for all $\alpha \in {\mathbb{R}}_{+}$ the dimension of the shrinking target set is zero.