American Institute of Mathematical Sciences

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

Contribution to the ergodic theory of robustly transitive maps

Cristina Lizana 1, , Vilton Pinheiro 2, and  Paulo Varandas 2,

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.
Keywords: endomorphisms, decay of correlations., expanding measures, invariant measures, ergodic measures, Robust transitivity.
Mathematics Subject Classification: Primary: 37A25, 37C40; Secondary: 37D2.
Citation: Cristina Lizana, Vilton Pinheiro, Paulo Varandas. Contribution to the ergodic theory of robustly transitive maps. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

C. Lizana, Robust Transitivity for Endomorphisms, Ph.D thesis, IMPA in Brazil, 2010. Google Scholar

[11]

C. Lizana and E. Pujals, Robust transitivity for endomorphisms, Ergodic Theory Dynam. Systems, 33 (2013), 1082-1114. doi: 10.1017/S0143385712000247.  Google Scholar

[12]

K. Moriyasu, The ergodic closing lemma for $C^1$ regular maps, Tokyo J. Math., 15 (1992), 171-183. doi: 10.3836/tjm/1270130259.  Google Scholar

[13]

V. Pinheiro, Expanding measures, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 889-939. doi: 10.1016/j.anihpc.2011.07.001.  Google Scholar

[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.  Google Scholar

[15]

P. Varandas, Statistical properties of generalized Viana maps, Dyn. Syst., 29 (2014), 167-189. doi: 10.1080/14689367.2013.868868.  Google Scholar

[16]

L. Wen, The $C^1$-Closing Lemma for nonsingular endomorphisms, Ergodic Theory Dynam. Systems, 11 (1991), 393-412. doi: 10.1017/S0143385700006210.  Google Scholar

[17]

L. Wen, A uniform $C^1$-connecting lemma, Discrete Contin. Dyn. Syst., 8 (2002), 257-265. doi: 10.3934/dcds.2002.8.257.  Google Scholar

[18]

L. S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

C. Lizana, Robust Transitivity for Endomorphisms, Ph.D thesis, IMPA in Brazil, 2010. Google Scholar

[11]

C. Lizana and E. Pujals, Robust transitivity for endomorphisms, Ergodic Theory Dynam. Systems, 33 (2013), 1082-1114. doi: 10.1017/S0143385712000247.  Google Scholar

[12]

K. Moriyasu, The ergodic closing lemma for $C^1$ regular maps, Tokyo J. Math., 15 (1992), 171-183. doi: 10.3836/tjm/1270130259.  Google Scholar

[13]

V. Pinheiro, Expanding measures, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 889-939. doi: 10.1016/j.anihpc.2011.07.001.  Google Scholar

[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.  Google Scholar

[15]

P. Varandas, Statistical properties of generalized Viana maps, Dyn. Syst., 29 (2014), 167-189. doi: 10.1080/14689367.2013.868868.  Google Scholar

[16]

L. Wen, The $C^1$-Closing Lemma for nonsingular endomorphisms, Ergodic Theory Dynam. Systems, 11 (1991), 393-412. doi: 10.1017/S0143385700006210.  Google Scholar

[17]

L. Wen, A uniform $C^1$-connecting lemma, Discrete Contin. Dyn. Syst., 8 (2002), 257-265. doi: 10.3934/dcds.2002.8.257.  Google Scholar

[18]

L. S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180.  Google Scholar

[1]

Fawwaz Batayneh, Cecilia González-Tokman. On the number of invariant measures for random expanding maps in higher dimensions. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5887-5914. doi: 10.3934/dcds.2021100
[2]

Oliver Jenkinson. Optimization and majorization of invariant measures. Electronic Research Announcements, 2007, 13: 1-12.

[3]

Siniša Slijepčević. Stability of invariant measures. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1345-1363. doi: 10.3934/dcds.2009.24.1345
[4]

Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185
[5]

Eugen Mihailescu. Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2485-2502. doi: 10.3934/dcds.2012.32.2485
[6]

Zhihong Xia. Hyperbolic invariant sets with positive measures. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 811-818. doi: 10.3934/dcds.2006.15.811
[7]

Marcus Pivato. Invariant measures for bipermutative cellular automata. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 723-736. doi: 10.3934/dcds.2005.12.723
[8]

Jiu Ding, Aihui Zhou. Absolutely continuous invariant measures for piecewise $C^2$ and expanding mappings in higher dimensions. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 451-458. doi: 10.3934/dcds.2000.6.451
[9]

Mrinal Kanti Roychowdhury. Quantization coefficients for ergodic measures on infinite symbolic space. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2829-2846. doi: 10.3934/dcds.2014.34.2829
[10]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457
[11]

Radu Saghin. On the number of ergodic minimizing measures for Lagrangian flows. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 501-507. doi: 10.3934/dcds.2007.17.501
[12]

Wen Huang, Leiye Xu, Shengnan Xu. Ergodic measures of intermediate entropy for affine transformations of nilmanifolds. Electronic Research Archive, 2021, 29 (4) : 2819-2827. doi: 10.3934/era.2021015
[13]

Émilie Chouzenoux, Henri Gérard, Jean-Christophe Pesquet. General risk measures for robust machine learning. Foundations of Data Science, 2019, 1 (3) : 249-269. doi: 10.3934/fods.2019011
[14]

Guizhen Cui, Yunping Jiang, Anthony Quas. Scaling functions and Gibbs measures and Teichmüller spaces of circle endomorphisms. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 535-552. doi: 10.3934/dcds.1999.5.535
[15]

Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597
[16]

Amir Mohammadi. Measures invariant under horospherical subgroups in positive characteristic. Journal of Modern Dynamics, 2011, 5 (2) : 237-254. doi: 10.3934/jmd.2011.5.237
[17]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017
[18]

Vasso Anagnostopoulou. Stochastic dominance for shift-invariant measures. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 667-682. doi: 10.3934/dcds.2019027
[19]

Gamaliel Blé. External arguments and invariant measures for the quadratic family. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 241-260. doi: 10.3934/dcds.2004.11.241
[20]

Zhang Chen, Xiliang Li, Bixiang Wang. Invariant measures of stochastic delay lattice systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3235-3269. doi: 10.3934/dcdsb.2020226

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (68)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences