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Local stability analysis of differential equations with state-dependent delay
Modified Schmidt games and non-dense forward orbits of partially hyperbolic systems
1. | School of Mathematical Sciences, Peking University, Beijing 100871, China |
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. |
show all references
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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