McMullen's game for equicontinuously-twisted badly approximable points in continued fractions and beta expansions

TL;DR

This paper extends McMullen's game results to equicontinuous sequences of functions in beta-transformations and Gauss maps, showing that certain badly approximable points form winning sets with full Hausdorff dimension.

Contribution

It generalizes known results from constant functions to all equicontinuous sequences in dynamical systems, establishing new winning set properties.

Findings

Collections are winning in McMullen's game for equicontinuous sequences.

Such sets have Hausdorff dimension 1.

Results apply to beta-transformations and Gauss maps.

Abstract

In a beta-transformation (for integer beta) or a Gauss map system, given a sequence of functions fn from [0,1] to itself, consider the collection of points in [0,1] whose nth iteration under the map is distanced away from its value under fn. It is well known that for constant sequences fn, such collections are always winning in McMullen's game and in particular they have Hausdorff dimension 1. We extend the results to all equicontinuous sequences of functions fn.