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

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 $\dim M$.

19 pages. Remark 4.10 is corrected. We have followed the proof scheme in \cite{Wu}. arXiv admin note: text overlap with arXiv:1311.5309