Modified Schmidt games and non-dense forward orbits of partially hyperbolic systems

TL;DR

This paper introduces modified Schmidt games tailored for partially hyperbolic systems to analyze the size and structure of points with non-dense forward orbits, showing these sets are large in terms of Hausdorff dimension.

Contribution

It generalizes previous results by developing a new game-theoretic approach for partially hyperbolic systems and establishing the large Hausdorff dimension of non-dense orbit sets.

Findings

Sets of points with non-dense forward orbits are winning sets in the modified Schmidt game.

These sets have Hausdorff dimension equal to the unstable manifold dimension.

The non-dense orbit sets intersecting open sets have full Hausdorff dimension in the manifold.

Abstract

Let $f: M \to M$ be a $C^{1+\theta}$-partially hyperbolic diffeomorphism. We introduce a type of modified Schmidt games which is induced by $f$ and played on any unstable manifold. Utilizing it we generalize some results of \cite{Wu} as follows. Consider a set of points with non-dense forward orbit: $$E(f, y) := \{ z\in M: y\notin \overline{\{f^k(z), k \in \mathbb{N}\}}\}$$ for some $y \in M$ and $$E_{x}(f, y) := E(f, y) \cap W^u(x)$$ for any $x\in M$. We show that $E_x(f,y)$ is a winning set for such modified Schmidt games played on $W^u(x)$, which implies that $E_x(f,y)$ has Hausdorff dimension equal to $\dim W^u(x)$. Then for any nonempty open set $V \subset M$ we show that $E(f, y) \cap V$ has full Hausdorff dimension equal to $\dim M$, by using a technique of constructing measures supported on $E(f, y)$ with lower pointwise dimension approximating $\dim M$.