On Shrinking Targets for Piecewise Expanding Interval Maps

For a map $T \colon [0,1] \to [0,1]$ with an invariant measure $μ$, we study, for a $μ$-typical $x$, the set of points $y$ such that the inequality $|T^n x - y| < r_n$ 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 $μ_ϕ$ is a Gibbs measure. In some cases we also show that the set has a large intersection property.

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