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A note on partially hyperbolic attractors: Entropy conjecture and SRB measures
Contribution to the ergodic theory of robustly transitive maps
1. | Departamento de Matemática, Facultad de Ciencias, La Hechicera, Universidad de los Andes Mérida, 5101 |
2. | Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil |
References:
[1] |
F. Abdenur, C. Bonatti, S. Crovisier, L. Díaz and L. Wen, Periodic points and homoclinic classes, Ergodic Theory Dynam. Systems, 27 (2007), 1-22.
doi: 10.1017/S0143385706000538. |
[2] |
J. F. Alves and V. Pinheiro, Gibbs-Markov structures and limit laws for partially hyperbolic attractors with mostly expanding central direction, Adv. Math., 223 (2010), 1706-1730.
doi: 10.1016/j.aim.2009.10.010. |
[3] |
C. Bonatti, L. Díaz and E. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. (2), 158 (2003), 355-418.
doi: 10.4007/annals.2003.158.355. |
[4] |
C. Bonatti, L. Díaz and R. Ures, Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms, J. Inst. Math. Jussieu, 1 (2002), 513-541.
doi: 10.1017/S1474748002000142. |
[5] |
J. Buzzi, T. Fisher, M. Sambarino and C. Vasquez, Maximal Entropy Measures for certain Partially Hyperbolic, Derived from Anosov systems, Ergodic Thery Dynam. Systems, 32 (2012), 63-79.
doi: 10.1017/S0143385710000854. |
[6] |
M. Carvalho, Sinai-Ruelle-Bowen measures for N-dimensional derived from Anosov diffeomorphisms, Ergodic Theory Dynam. Systems, 13 (1993), 21-44.
doi: 10.1017/S0143385700007185. |
[7] |
A. Castro, Fast mixing for attractors with a mostly contracting central direction, Ergodic Theory Dynam. Systems, 24 (2004), 17-44.
doi: 10.1017/S0143385703000294. |
[8] |
A. Castro, New criteria for hyperbolicity based on periodic points, Bull. Braz. Math. Soc., 42 (2011), 455-483.
doi: 10.1007/s00574-011-0025-4. |
[9] |
S. Hayashi, Connecting invariant manifolds and the solution of the $C^1-$stability and $\Omega-$stability conjectures for flows, Ann. of Math.(2), 145 (1997), 81-137.
doi: 10.2307/2951824. |
[10] |
C. Lizana, Robust Transitivity for Endomorphisms, Ph.D thesis, IMPA in Brazil, 2010. |
[11] |
C. Lizana and E. Pujals, Robust transitivity for endomorphisms, Ergodic Theory Dynam. Systems, 33 (2013), 1082-1114.
doi: 10.1017/S0143385712000247. |
[12] |
K. Moriyasu, The ergodic closing lemma for $C^1$ regular maps, Tokyo J. Math., 15 (1992), 171-183.
doi: 10.3836/tjm/1270130259. |
[13] |
V. Pinheiro, Expanding measures, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 889-939.
doi: 10.1016/j.anihpc.2011.07.001. |
[14] |
R. Ures, Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part, Proc. Amer. Math. Soc., 140 (2012), 1973-1985.
doi: 10.1090/S0002-9939-2011-11040-2. |
[15] |
P. Varandas, Statistical properties of generalized Viana maps, Dyn. Syst., 29 (2014), 167-189.
doi: 10.1080/14689367.2013.868868. |
[16] |
L. Wen, The $C^1$-Closing Lemma for nonsingular endomorphisms, Ergodic Theory Dynam. Systems, 11 (1991), 393-412.
doi: 10.1017/S0143385700006210. |
[17] |
L. Wen, A uniform $C^1$-connecting lemma, Discrete Contin. Dyn. Syst., 8 (2002), 257-265.
doi: 10.3934/dcds.2002.8.257. |
[18] |
L. S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188.
doi: 10.1007/BF02808180. |
show all references
References:
[1] |
F. Abdenur, C. Bonatti, S. Crovisier, L. Díaz and L. Wen, Periodic points and homoclinic classes, Ergodic Theory Dynam. Systems, 27 (2007), 1-22.
doi: 10.1017/S0143385706000538. |
[2] |
J. F. Alves and V. Pinheiro, Gibbs-Markov structures and limit laws for partially hyperbolic attractors with mostly expanding central direction, Adv. Math., 223 (2010), 1706-1730.
doi: 10.1016/j.aim.2009.10.010. |
[3] |
C. Bonatti, L. Díaz and E. Pujals, A $C^1$-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of Math. (2), 158 (2003), 355-418.
doi: 10.4007/annals.2003.158.355. |
[4] |
C. Bonatti, L. Díaz and R. Ures, Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms, J. Inst. Math. Jussieu, 1 (2002), 513-541.
doi: 10.1017/S1474748002000142. |
[5] |
J. Buzzi, T. Fisher, M. Sambarino and C. Vasquez, Maximal Entropy Measures for certain Partially Hyperbolic, Derived from Anosov systems, Ergodic Thery Dynam. Systems, 32 (2012), 63-79.
doi: 10.1017/S0143385710000854. |
[6] |
M. Carvalho, Sinai-Ruelle-Bowen measures for N-dimensional derived from Anosov diffeomorphisms, Ergodic Theory Dynam. Systems, 13 (1993), 21-44.
doi: 10.1017/S0143385700007185. |
[7] |
A. Castro, Fast mixing for attractors with a mostly contracting central direction, Ergodic Theory Dynam. Systems, 24 (2004), 17-44.
doi: 10.1017/S0143385703000294. |
[8] |
A. Castro, New criteria for hyperbolicity based on periodic points, Bull. Braz. Math. Soc., 42 (2011), 455-483.
doi: 10.1007/s00574-011-0025-4. |
[9] |
S. Hayashi, Connecting invariant manifolds and the solution of the $C^1-$stability and $\Omega-$stability conjectures for flows, Ann. of Math.(2), 145 (1997), 81-137.
doi: 10.2307/2951824. |
[10] |
C. Lizana, Robust Transitivity for Endomorphisms, Ph.D thesis, IMPA in Brazil, 2010. |
[11] |
C. Lizana and E. Pujals, Robust transitivity for endomorphisms, Ergodic Theory Dynam. Systems, 33 (2013), 1082-1114.
doi: 10.1017/S0143385712000247. |
[12] |
K. Moriyasu, The ergodic closing lemma for $C^1$ regular maps, Tokyo J. Math., 15 (1992), 171-183.
doi: 10.3836/tjm/1270130259. |
[13] |
V. Pinheiro, Expanding measures, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 889-939.
doi: 10.1016/j.anihpc.2011.07.001. |
[14] |
R. Ures, Intrinsic ergodicity of partially hyperbolic diffeomorphisms with a hyperbolic linear part, Proc. Amer. Math. Soc., 140 (2012), 1973-1985.
doi: 10.1090/S0002-9939-2011-11040-2. |
[15] |
P. Varandas, Statistical properties of generalized Viana maps, Dyn. Syst., 29 (2014), 167-189.
doi: 10.1080/14689367.2013.868868. |
[16] |
L. Wen, The $C^1$-Closing Lemma for nonsingular endomorphisms, Ergodic Theory Dynam. Systems, 11 (1991), 393-412.
doi: 10.1017/S0143385700006210. |
[17] |
L. Wen, A uniform $C^1$-connecting lemma, Discrete Contin. Dyn. Syst., 8 (2002), 257-265.
doi: 10.3934/dcds.2002.8.257. |
[18] |
L. S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188.
doi: 10.1007/BF02808180. |
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