Schmidt's Game and Nonuniformly Expanding Interval 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.

16 pages, 6 figures. This version corrects typos, improves exposition in the main proof, and adds a remark concerning the assumption we place on the maps under consideration