Shrinking targets in parametrised families

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.

29 pages, 0 figures. Correction of the upper bound on the Hausdorff dimension, and some reorganisation of the paper

References in corpus (1)

• Uniform Diophantine approximation related to $b$-ary and $\beta$-expansions

Cited by in corpus (2)

• Shrinking targets and eventually always hitting points for interval maps
• The Dolgopyat inequality in \BV\ for non-Markov maps