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

Published online by Cambridge University Press:  11 February 2015

WEISHENG WU
WEISHENG WU*
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA email [email protected]
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Abstract

Let $f:M\rightarrow M$ be a partially hyperbolic diffeomorphism with conformality on unstable manifolds. 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 $y\in M$. Define $E_{x}(f,y):=E(f,y)\cap W^{u}(x)$ for any $x\in M$. Following a method of Broderick, Fishman and Kleinbock [Schmidt’s game, fractals, and orbits of toral endomorphisms. Ergod. Th. & Dynam. Sys.31 (2011), 1095–1107], we show that $E_{x}(f,y)$ is a winning set for Schmidt games played on $W^{u}(x)$ which implies that $E_{x}(f,y)$ has Hausdorff dimension equal to $\dim W^{u}(x)$. Furthermore, we show that for any non-empty 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 $M$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 5 , August 2016 , pp. 1656 - 1678
Copyright
© Cambridge University Press, 2015 

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