# American Institute of Mathematical Sciences

January  2015, 35(1): 353-365. doi: 10.3934/dcds.2015.35.353

## 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

Received  January 2014 Revised  April 2014 Published  August 2014

In this article we intend to contribute in the understanding of the ergodic properties of the set of robustly transitive local diffeomorphisms on a compact manifold without boundary. We prove that $C^1$ generic robustly transitive local diffeomorphisms have a residual subset of points with dense pre-orbits. Moreover, $C^1$ generically in the space of local diffeomorphisms with no splitting and all points with dense pre-orbits, there are uncountably many ergodic expanding invariant measures with full support and exhibiting exponential decay of correlations. In particular, these results hold for an important class of robustly transitive maps.
Citation: Cristina Lizana, Vilton Pinheiro, Paulo Varandas. Contribution to the ergodic theory of robustly transitive maps. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 353-365. doi: 10.3934/dcds.2015.35.353
##### 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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##### 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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