On Shrinking Targets for Piecewise Expanding Interval Maps

For a map $T\text{\hspace{0.17em}⁣}:[0,1]\to [0,1]$ with an invariant measure $\mu$, we study, for a $\mu$-typical $x$, the set of points $y$ such that the inequality $\mathrm{\mid }{T}^{n}x-y\mathrm{\mid }<r_n$ is satisfied for infinitely many $n$. We give a formula for the Hausdorff dimension of this set, under the assumption that $T$ is piecewise expanding and ${\mu }_{\varphi }$ is a Gibbs measure. In some cases we also show that the set has a large intersection property.