# 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 and Continuous Dynamical Systems, 2016, 36 (6) : 3463-3481. doi: 10.3934/dcds.2016.36.3463
##### References:
 [1] J. An, Two dimensional badly approximable vectors and Schmidt's game,, Duke Mathematical Journal, (). [2] C. S. Aravinda and E. Leuzinger, Bounded geodesics in rank-1 locally symmetric spaces, Ergodic Theory and Dynamical Systems, 15 (1995), 813-820. doi: 10.1017/S0143385700009640. [3] M. Brin and Ja. B. Pesin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212. [4] R. Broderick, L. Fishman and D. Y. Kleinbock, Schmidt's game, fractals, and orbits of toral endomorphisms, Ergodic Theory Dynam. Systems, 31 (2011), 1095-1107. doi: 10.1017/S0143385710000374. [5] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Annals of Math., 171 (2010), 451-489. doi: 10.4007/annals.2010.171.451. [6] S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. reine angew. Math., 359 (1985), 55-89. doi: 10.1515/crll.1985.359.55. [7] S. G. Dani, Bounded orbits of flows on homogeneous spaces, Commentarii Mathematici Helvetici, 61 (1986), 636-660. doi: 10.1007/BF02621936. [8] S. G. Dani, On orbits of endomorphisms of tori and the Schmidt game, Ergodic Theory and Dynamical Systems, 8 (1988), 523-529. doi: 10.1017/S0143385700004673. [9] S. G. Dani and H. Shah, Badly approximable numbers and vectors in Cantor-like sets, Proceedings of the American Mathematical Society, 140 (2012), 2575-2587. doi: 10.1090/S0002-9939-2011-11105-5. [10] D. Dolgopyat, Bounded orbits of Anosov flows, Duke Mathematical Journal, 87 (1997), 87-114. doi: 10.1215/S0012-7094-97-08704-4. [11] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, 2007. doi: 10.1002/0470013850. [12] J. M. Franks, Invariant sets of hyperbolic toral automorphisms, American Journal of Mathematics, 99 (1977), 1089-1095. doi: 10.2307/2374001. [13] D. Y. Kleinbock and G. A. Margulis, {Bounded orbits of nonquasiunipotent flows on homogeneous spaces, American Mathematical Society Translations, 171 (1996), 141-172. [14] D. Y. Kleinbock and B. Weiss, Modified Schmidt games and Diophantine approximation with weights, Advances in Mathematics, 223 (2010), 1276-1298. doi: 10.1016/j.aim.2009.09.018. [15] \bysame, Modified Schmidt games and a conjecture of Margulis, Journal of Modern Dynamics, 7 (2013), 429-460. doi: 10.3934/jmd.2013.7.429. [16] R. Mañé, Orbits of paths under hyperbolic toral automorphisms, Proceedings of the American Mathematical Society, 73 (1979), 121-125. doi: 10.1090/S0002-9939-1979-0512072-3. [17] C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Transactions of the American Mathematical Society, 300 (1987), 329-342. doi: 10.1090/S0002-9947-1987-0871679-3. [18] F. Przytycki, Construction of invariant sets for Anosov diffeomorphisms and hyperbolic attractors, Studia Mathematica, 68 (1980), 199-213. [19] C. Pugh and M. Shub, Ergodicity of Anosov actions, Inventiones mathematicae, 15 (1972), 1-23. doi: 10.1007/BF01418639. [20] F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1d-center bundle, Inventiones mathematicae, 172 (2008), 353-381. doi: 10.1007/s00222-007-0100-z. [21] W. M. Schmidt, On badly approximable numbers and certain games, Transactions of the American Mathematical Society, 123 (1966), 178-199. doi: 10.1090/S0002-9947-1966-0195595-4. [22] W. M. Schmidt, Badly approximable systems of linear forms, Journal of Number Theory, 1 (1969), 139-154. doi: 10.1016/0022-314X(69)90032-8. [23] J. Tseng, Schmidt games and Markov partitions, Nonlinearity, 22 (2009), 525-543. doi: 10.1088/0951-7715/22/3/001. [24] M. Urbański, The Hausdorff dimension of the set of points with nondense orbit under a hyperbolic dynamical system, Nonlinearity, 4 (1991), 385-397. doi: 10.1088/0951-7715/4/2/009. [25] W. Wu, Schmidt games and non-dense forward orbits of certain partially hyperbolic systems,, Ergodic Theory and Dynamical Systems, ().  doi: 10.1017/etds.2014.136.

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##### References:
 [1] J. An, Two dimensional badly approximable vectors and Schmidt's game,, Duke Mathematical Journal, (). [2] C. S. Aravinda and E. Leuzinger, Bounded geodesics in rank-1 locally symmetric spaces, Ergodic Theory and Dynamical Systems, 15 (1995), 813-820. doi: 10.1017/S0143385700009640. [3] M. Brin and Ja. B. Pesin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212. [4] R. Broderick, L. Fishman and D. Y. Kleinbock, Schmidt's game, fractals, and orbits of toral endomorphisms, Ergodic Theory Dynam. Systems, 31 (2011), 1095-1107. doi: 10.1017/S0143385710000374. [5] K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Annals of Math., 171 (2010), 451-489. doi: 10.4007/annals.2010.171.451. [6] S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, J. reine angew. Math., 359 (1985), 55-89. doi: 10.1515/crll.1985.359.55. [7] S. G. Dani, Bounded orbits of flows on homogeneous spaces, Commentarii Mathematici Helvetici, 61 (1986), 636-660. doi: 10.1007/BF02621936. [8] S. G. Dani, On orbits of endomorphisms of tori and the Schmidt game, Ergodic Theory and Dynamical Systems, 8 (1988), 523-529. doi: 10.1017/S0143385700004673. [9] S. G. Dani and H. Shah, Badly approximable numbers and vectors in Cantor-like sets, Proceedings of the American Mathematical Society, 140 (2012), 2575-2587. doi: 10.1090/S0002-9939-2011-11105-5. [10] D. Dolgopyat, Bounded orbits of Anosov flows, Duke Mathematical Journal, 87 (1997), 87-114. doi: 10.1215/S0012-7094-97-08704-4. [11] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, 2007. doi: 10.1002/0470013850. [12] J. M. Franks, Invariant sets of hyperbolic toral automorphisms, American Journal of Mathematics, 99 (1977), 1089-1095. doi: 10.2307/2374001. [13] D. Y. Kleinbock and G. A. Margulis, {Bounded orbits of nonquasiunipotent flows on homogeneous spaces, American Mathematical Society Translations, 171 (1996), 141-172. [14] D. Y. Kleinbock and B. Weiss, Modified Schmidt games and Diophantine approximation with weights, Advances in Mathematics, 223 (2010), 1276-1298. doi: 10.1016/j.aim.2009.09.018. [15] \bysame, Modified Schmidt games and a conjecture of Margulis, Journal of Modern Dynamics, 7 (2013), 429-460. doi: 10.3934/jmd.2013.7.429. [16] R. Mañé, Orbits of paths under hyperbolic toral automorphisms, Proceedings of the American Mathematical Society, 73 (1979), 121-125. doi: 10.1090/S0002-9939-1979-0512072-3. [17] C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Transactions of the American Mathematical Society, 300 (1987), 329-342. doi: 10.1090/S0002-9947-1987-0871679-3. [18] F. Przytycki, Construction of invariant sets for Anosov diffeomorphisms and hyperbolic attractors, Studia Mathematica, 68 (1980), 199-213. [19] C. Pugh and M. Shub, Ergodicity of Anosov actions, Inventiones mathematicae, 15 (1972), 1-23. doi: 10.1007/BF01418639. [20] F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1d-center bundle, Inventiones mathematicae, 172 (2008), 353-381. doi: 10.1007/s00222-007-0100-z. [21] W. M. Schmidt, On badly approximable numbers and certain games, Transactions of the American Mathematical Society, 123 (1966), 178-199. doi: 10.1090/S0002-9947-1966-0195595-4. [22] W. M. Schmidt, Badly approximable systems of linear forms, Journal of Number Theory, 1 (1969), 139-154. doi: 10.1016/0022-314X(69)90032-8. [23] J. Tseng, Schmidt games and Markov partitions, Nonlinearity, 22 (2009), 525-543. doi: 10.1088/0951-7715/22/3/001. [24] M. Urbański, The Hausdorff dimension of the set of points with nondense orbit under a hyperbolic dynamical system, Nonlinearity, 4 (1991), 385-397. doi: 10.1088/0951-7715/4/2/009. [25] W. Wu, Schmidt games and non-dense forward orbits of certain partially hyperbolic systems,, Ergodic Theory and Dynamical Systems, ().  doi: 10.1017/etds.2014.136.
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