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

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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

Abstract
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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 $\dim M$.

Keywords: partially hyperbolic diffeomorphism, non-dense orbits, Schmidt games, Hausdorff dimension..

Mathematics Subject Classification: Primary: 37D30, 37C45; Secondary: 28A9.

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

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References:

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

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