# American Institute of Mathematical Sciences

June  2016, 36(6): 3463-3481. doi: 10.3934/dcds.2016.36.3463

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

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

Received  April 2015 Revised  October 2015 Published  December 2015

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 $\dim M$.
Citation: Weisheng Wu. Modified Schmidt games and non-dense forward orbits of partially hyperbolic systems. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3463-3481. doi: 10.3934/dcds.2016.36.3463
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##### References:
 [1] Vanderlei Horita, Marcelo Viana. Hausdorff dimension for non-hyperbolic repellers II: DA diffeomorphisms. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1125-1152. doi: 10.3934/dcds.2005.13.1125 [2] Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020 [3] Thomas Barthelmé, Andrey Gogolev. Centralizers of partially hyperbolic diffeomorphisms in dimension 3. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4477-4484. doi: 10.3934/dcds.2021044 [4] Dmitry Kleinbock, Barak Weiss. Modified Schmidt games and a conjecture of Margulis. Journal of Modern Dynamics, 2013, 7 (3) : 429-460. doi: 10.3934/jmd.2013.7.429 [5] Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993 [6] Todd Young. Partially hyperbolic sets from a co-dimension one bifurcation. Discrete & Continuous Dynamical Systems, 1995, 1 (2) : 253-275. doi: 10.3934/dcds.1995.1.253 [7] Yan Huang. On Hausdorff dimension of the set of non-ergodic directions of two-genus double cover of tori. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2395-2409. doi: 10.3934/dcds.2018099 [8] Kei Irie. Dense existence of periodic Reeb orbits and ECH spectral invariants. Journal of Modern Dynamics, 2015, 9: 357-363. doi: 10.3934/jmd.2015.9.357 [9] Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591 [10] Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503 [11] Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405 [12] Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118. [13] Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457 [14] Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293 [15] Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435 [16] Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 505-512. doi: 10.3934/dcds.2006.16.505 [17] Rafael Potrie. Partially hyperbolic diffeomorphisms with a trapping property. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 5037-5054. doi: 10.3934/dcds.2015.35.5037 [18] Lorenzo J. Díaz, Todd Fisher. Symbolic extensions and partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1419-1441. doi: 10.3934/dcds.2011.29.1419 [19] Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Journal of Modern Dynamics, 2010, 4 (2) : 271-327. doi: 10.3934/jmd.2010.4.271 [20] Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68

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