Schmidt games and Markov partitions

TL;DR

This paper corrects and strengthens a theorem about the Hausdorff dimension of points with nondense orbits under an expanding map, and extends it using Schmidt games to show certain sets are winning in dimension one.

Contribution

It corrects a gap in a key theorem and generalizes the result to show that sets of points missing a specific orbit point are winning sets in Schmidt games.

Findings

The set of points with nondense orbits has full Hausdorff dimension.

The set of points whose orbits miss a given point is winning in Schmidt games.

The correction allows a stronger version of the original theorem.

Abstract

Let T be a C^2-expanding self-map of a compact, connected, smooth, Riemannian manifold M. We correct a minor gap in the proof of a theorem from the literature: the set of points whose forward orbits are nondense has full Hausdorff dimension. Our correction allows us to strengthen the theorem. Combining the correction with Schmidt games, we generalize the theorem in dimension one: given a point x in M, the set of points whose forward orbit closures miss x is a winning set.