Schmidt Games and Nondense forward Orbits of certain Partially Hyperbolic Systems

TL;DR

This paper proves that the set of points with nondense forward orbits in certain partially hyperbolic systems is large in terms of Hausdorff dimension, using Schmidt games and measure construction techniques.

Contribution

It establishes that these nondense orbit sets are winning in Schmidt games and have full Hausdorff dimension, extending previous results to a broader class of systems.

Findings

Sets of points with nondense orbits are winning in Schmidt games.

These sets have full Hausdorff dimension equal to the unstable manifold or the entire manifold.

The results extend to points avoiding countable subsets of the manifold.

Abstract

Let $f: M \to M$ be a partially hyperbolic diffeomorphism with conformality on unstable manifolds. Consider a set of points with nondense forward orbit: $E(f, y) := \{ z\in M: y\notin \overline{\{f^k(z), k \in \mathbb{N}\}}\}$ for some $y \in M$. Define $E_{x}(f, y) := E(f, y) \cap W^u(x)$ for any $x\in M$. Following a method of Broderick-Fishman-Kleinbock, we show that $E_x(f,y)$ is a winning set of Schmidt games played on $W^u(x)$ which implies that $E_x(f,y)$ has full Hausdorff dimension equal to $\dim W^u(x)$. Furthermore we show that for any nonempty open set $V \subset M$, $E(f, y) \cap V$ has full Hausdorff dimension equal to $\dim M$, by constructing measures supported on $E(f, y)\cap V$ with lower pointwise dimension converging to $\dim M$ and with conditional measures supported on $E_x(f,y)\cap V$. The results can be extended to the set of points with forward orbit staying away from a countable subset of $M$.