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Classifying GL$(n,\mathbb{Z})$-orbits of points and rational subspaces
Dominated splitting, partial hyperbolicity and positive entropy
1. | Instituto de Matemática y Estadística Prof. Rafael Laguardia (IMERL), Facultad de Ingeniería, Universidad de la República |
2. | School of Mathematical Sciences, Fudan University, Shanghai 200433, China |
References:
[1] |
L. Barreira and Y. B. Pesin, Nonuniform Hyperbolicity, Cambridge Univ. Press, Cambridge, 2007.
doi: 10.1017/CBO9781107326026. |
[2] |
J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic systems, Ann. of Math., 161 (2005), 1423-1485.
doi: 10.4007/annals.2005.161.1423. |
[3] |
C. Bonatti, L. Diaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, III. Springer-Verlag, Berlin, 2005. |
[4] |
R. Bowen, Periodic point and measures for axiom-A-diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397. |
[5] |
R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.
doi: 10.1090/S0002-9947-1973-0338317-X. |
[6] |
E. Catsigeras, M. Cerminara and H. Enrich, Pesin's entropy formula for $C^1$ diffeomorphisms with dominated splitting, Ergodic Theory and Dynamical Systems, 35 (2015), 737-761.
doi: 10.1017/etds.2013.93. |
[7] |
E. Catsigeras and H. Enrich, SRB-like measures for $C^0$ dynamics, Bull. Pol. Acad. Sci. Math., 59 (2011), 151-164.
doi: 10.4064/ba59-2-5. |
[8] |
M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on the Compact Space, Lecture Notes in Mathematics, 527. |
[9] |
L. J. Díaz, T. Fisher, M. J. Pacifico and J. L. Vieitez, Symbolic extensions for partially hyperbolic diffeomorphisms, Discrete and Continuous Dynamical Systems, 32 (2012), 4195-4207.
doi: 10.3934/dcds.2012.32.4195. |
[10] |
N. Gourmelon, Addapted metrics for dominated splitting, Ergod. Th. and Dyn. Sys., 27 (2007), 1839-1849.
doi: 10.1017/S0143385707000272. |
[11] |
N. Gourmelon and R. Potrie, Projectively Anosov Diffeomorphisms of Surfaces, Preprint to appear. Personal communication, 2015. |
[12] |
G. Liao, M. Viana and J. Yang, The entropy conjecture for diffeomorphisms away from tangencies, Journal of the European Mathematical Society, 15 (2013), 2043-2060.
doi: 10.4171/JEMS/413. |
[13] |
V. I. Oseledec, Multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 179-210; translated from Russian. |
[14] |
M. J. Pacifico and J. L. Vieitez, Entropy-expansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut., 21 (2008), 293-317.
doi: 10.5209/rev_REMA.2008.v21.n2.16370. |
[15] |
H. Qiu, Existence and uniqueness of SRB measure on $C^1$ generic hyperbolic attractors, Commun. Math. Phys., 302 (2011), 345-357.
doi: 10.1007/s00220-010-1160-2. |
[16] |
R. Saghin, W. Sun and E. Vargas, Ergodic properties of time-changes for flows, preprint. |
[17] |
N. Sumi, P. Varandas and K. Yamamoto, Partial hyperbolicity and specification, Proc. Amer. Math. Soc., 144 (2016), 1161-1170.
doi: 10.1090/proc/12830. |
[18] |
W. Sun and X. Tian, Dominated splitting and Pesin's entropy formula, Discrete and Continuous Dynamical Systems, 32 (2012), 1421-1434. |
[19] |
P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982. |
[20] |
show all references
References:
[1] |
L. Barreira and Y. B. Pesin, Nonuniform Hyperbolicity, Cambridge Univ. Press, Cambridge, 2007.
doi: 10.1017/CBO9781107326026. |
[2] |
J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic systems, Ann. of Math., 161 (2005), 1423-1485.
doi: 10.4007/annals.2005.161.1423. |
[3] |
C. Bonatti, L. Diaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, III. Springer-Verlag, Berlin, 2005. |
[4] |
R. Bowen, Periodic point and measures for axiom-A-diffeomorphisms, Trans. Amer. Math. Soc., 154 (1971), 377-397. |
[5] |
R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136.
doi: 10.1090/S0002-9947-1973-0338317-X. |
[6] |
E. Catsigeras, M. Cerminara and H. Enrich, Pesin's entropy formula for $C^1$ diffeomorphisms with dominated splitting, Ergodic Theory and Dynamical Systems, 35 (2015), 737-761.
doi: 10.1017/etds.2013.93. |
[7] |
E. Catsigeras and H. Enrich, SRB-like measures for $C^0$ dynamics, Bull. Pol. Acad. Sci. Math., 59 (2011), 151-164.
doi: 10.4064/ba59-2-5. |
[8] |
M. Denker, C. Grillenberger and K. Sigmund, Ergodic Theory on the Compact Space, Lecture Notes in Mathematics, 527. |
[9] |
L. J. Díaz, T. Fisher, M. J. Pacifico and J. L. Vieitez, Symbolic extensions for partially hyperbolic diffeomorphisms, Discrete and Continuous Dynamical Systems, 32 (2012), 4195-4207.
doi: 10.3934/dcds.2012.32.4195. |
[10] |
N. Gourmelon, Addapted metrics for dominated splitting, Ergod. Th. and Dyn. Sys., 27 (2007), 1839-1849.
doi: 10.1017/S0143385707000272. |
[11] |
N. Gourmelon and R. Potrie, Projectively Anosov Diffeomorphisms of Surfaces, Preprint to appear. Personal communication, 2015. |
[12] |
G. Liao, M. Viana and J. Yang, The entropy conjecture for diffeomorphisms away from tangencies, Journal of the European Mathematical Society, 15 (2013), 2043-2060.
doi: 10.4171/JEMS/413. |
[13] |
V. I. Oseledec, Multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 179-210; translated from Russian. |
[14] |
M. J. Pacifico and J. L. Vieitez, Entropy-expansiveness and domination for surface diffeomorphisms, Rev. Mat. Complut., 21 (2008), 293-317.
doi: 10.5209/rev_REMA.2008.v21.n2.16370. |
[15] |
H. Qiu, Existence and uniqueness of SRB measure on $C^1$ generic hyperbolic attractors, Commun. Math. Phys., 302 (2011), 345-357.
doi: 10.1007/s00220-010-1160-2. |
[16] |
R. Saghin, W. Sun and E. Vargas, Ergodic properties of time-changes for flows, preprint. |
[17] |
N. Sumi, P. Varandas and K. Yamamoto, Partial hyperbolicity and specification, Proc. Amer. Math. Soc., 144 (2016), 1161-1170.
doi: 10.1090/proc/12830. |
[18] |
W. Sun and X. Tian, Dominated splitting and Pesin's entropy formula, Discrete and Continuous Dynamical Systems, 32 (2012), 1421-1434. |
[19] |
P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York-Berlin, 1982. |
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