On Shrinking Targets for Piecewise Expanding Interval Maps

TL;DR

This paper investigates the Hausdorff dimension of points that are frequently close to a typical orbit in piecewise expanding interval maps, providing a formula under Gibbs measure assumptions and exploring large intersection properties.

Contribution

It offers a new formula for the Hausdorff dimension of shrinking target sets in piecewise expanding maps with Gibbs measures, extending previous results in dynamical systems.

Findings

Derived a formula for Hausdorff dimension of shrinking target sets

Established large intersection properties for these sets

Applied results to piecewise expanding interval maps with Gibbs measures

Abstract

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