Schmidt's Game and Nonuniformly Expanding Interval Maps

TL;DR

This paper investigates the properties of certain nonuniformly expanding interval maps, proving that the set of points avoiding a specific interval is large in the sense of Schmidt's game, extending known results to more complex systems.

Contribution

It extends the theory of Schmidt's game to nonuniformly expanding maps, handling challenges like infinite distortion and unbounded geometry.

Findings

The set of points missing an interval with endpoint 0 is strong winning.

Strong winning sets are dense and have full Hausdorff dimension.

Results generalize previous work from expanding to nonuniformly expanding maps.

Abstract

We study Manneville-Pomeau maps on the unit interval and prove that the set of points whose forward orbits miss an interval with left endpoint 0 is strong winning for Schmidt's game. Strong winning sets are dense, have full Hausdorff dimension, and satisfy a countable intersection property. Similar results were known for certain expanding maps, but these did not address the nonuniformly expanding case. Our analysis is complicated by the presence of infinite distortion and unbounded geometry.