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

Weisheng  Wu

doi:10.3934/dcds.2016.36.3463

Article Contents

2016, Volume 36, Issue 6: 3463-3481. Doi: 10.3934/dcds.2016.36.3463

This issue Previous Article Local stability analysis of differential equations with state-dependent delay Next Article

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

Weisheng Wu^1,

1.
School of Mathematical Sciences, Peking University, Beijing 100871

Received: March 31, 2015

Revised: September 30, 2015

Published: May 2017

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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 [25] 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 $\dim M$.

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Mathematics Subject Classification: Primary: 37D30, 37C45; Secondary: 28A99.

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