Shrinking targets in parametrised families

TL;DR

This paper investigates the Hausdorff dimension of parameter sets in families of piecewise expanding maps where a fixed point's orbit hits shrinking targets, extending previous results to more general non-linear maps.

Contribution

It generalizes existing results on $eta$-transformations to broader non-linear families of maps, providing estimates and calculations for the Hausdorff dimension of relevant parameter sets.

Findings

Derived bounds for Hausdorff dimension of parameter sets

Extended results to non-linear families of maps

Utilized Schnellmann's typicality theorem in proofs

Abstract

We consider certain parametrised families of piecewise expanding maps on the interval, and estimate and sometimes calculate the Hausdorff dimension of the set of parameters for which the orbit of a fixed point has a certain shrinking target property. This generalises several similar results for $\beta$-transformations to more general non-linear families. The proofs are based on a result by Schnellmann on typicality in parametrised families.