Schmidt Games, Nonuniform Hyperbolicity, and Topological Entropy

In this work we accomplish several goals. First, we show how a geometric game introduced by Schmidt can be used to estimate various notions of the size of some interesting sets in dynamical systems. Specifically, we analyze so-called exceptional sets (sets of points whose orbit closures miss a prescribed point) arising from a class of nonuniformly expanding circle maps and show that, under reasonable conditions, such sets are winning for Schmidt's game. This implies that these sets are quite large in a certain sense, having full Hausdorff dimension despite being Lebesgue-null. Second, we show how a dynamical variation of Schmidt's game introduced by Weisheng Wu may be employed to measure other Caratheodory dimension characteristics besides Hausdorff dimensions, such as topological entropy. In particular, we modify and further develop the theory of this dynamical game, showing that even if a set is not winning, one can still derive lower bounds for the entropy of a given map on the set. We also prove that the winning property of the new game is an invariant of topological conjugacy, and that the winning property is independent of the scale of the Bowen balls used in the game.