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April  2020, 25(4): 1361-1382. doi: 10.3934/dcdsb.2019231

Ergodicity of non-autonomous discrete systems with non-uniform expansion

Pablo G. Barrientos 1,, and  Abbas Fakhari 2,

1. 

Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói Brasil
2. 

Department of Mathematics, Faculty of Mathematical sciences, Shahid Beheshti University G.C, Tehran, Iran

* Corresponding author: Pablo Barrientos

Received  May 2018 Revised  November 2018 Published  April 2020 Early access  November 2019

We study the ergodicity of non-autonomous discrete dynamical systems with non-uniform expansion. As an application we get that any uniformly expanding finitely generated semigroup action of $ C^{1+\alpha} $ local diffeomorphisms of a compact manifold is ergodic with respect to the Lebesgue measure. Moreover, we will also prove that every exact non-uniform expandable finitely generated semigroup action of conformal $ C^{1+\alpha} $ local diffeomorphisms of a compact manifold is Lebesgue ergodic.

Keywords: Ergodicity, non-autonomoous discrete system, non-uniform expansion.
Mathematics Subject Classification: 37C40, 37C60.
Citation: Pablo G. Barrientos, Abbas Fakhari. Ergodicity of non-autonomous discrete systems with non-uniform expansion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1361-1382. doi: 10.3934/dcdsb.2019231
References:
[1]

J. F. Alves, SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Scient. Éc. Norm. Sup., 33 (2000), 1-32.  doi: 10.1016/S0012-9593(00)00101-4.  Google Scholar

[2]

J. F. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398.  doi: 10.1007/s002220000057.  Google Scholar

[3]

J. F. Alves, S. Luzzatto and V. Pinheiro, Markov structures and decay of correlations for non-uniformly expanding dynamical systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 817-839.  doi: 10.1016/j.anihpc.2004.12.002.  Google Scholar

[4]

J. F. Alves and H. Vilarinho, Strong stochastic stability for non-uniformly expanding maps, Ergod. Th. Dynam. Sys., 33 (2013), 647-692.  doi: 10.1017/S0143385712000077.  Google Scholar

[5]

P. G. Barrientos, A. Fakhari, D. Malicet and A. Sarizadeh, Expanding actions: Minimality and ergodicity, Stoch. Dyn., 17 (2017), 1750031, 20 pp. doi: 10.1142/S0219493717500319.  Google Scholar

[6]

R. Bowen, A horseshoe with positive measure, Invent. Math., 29 (1975), 203-204.  doi: 10.1007/BF01389849.  Google Scholar

[7]

Y. L. Cao, S. Luzzatto and I. Rios, Uniform hyperbolicity for random maps with positive Lyapunov exponents, Proc. Amer. Math. Soc., 136 (2008), 3591-3600.  doi: 10.1090/S0002-9939-08-09347-7.  Google Scholar

[8]

M. Denker and M. Urbański, On the existence of conformal measure, Trans. Amer. Math. Soc., 328 (1991), 563-587.  doi: 10.1090/S0002-9947-1991-1014246-4.  Google Scholar

[9]

E. Durand-Cartagena and J. A. Jaramillo, Pointwise Lipschitz functions on metric spaces, J. Math. Anal. Appl., 363 (2010), 525-548.  doi: 10.1016/j.jmaa.2009.09.039.  Google Scholar

[10]

A. Ehsani, F.-H. Ghane and M. Zaj, On Ergodicity of Mostly Expanding Semi-Group Actions, 2015, arXiv: 1505.03367. Google Scholar

[11]

M. Gharaei and A. J. Homburg, Random interval diffeomorphisms, Discrete & Continuous Dynamical Systems-S, 10 (2017), 241-272.  doi: 10.3934/dcdss.2017012.  Google Scholar

[12] J. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson, Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients, New Mathematical Monographs, 27. Cambridge University Press, Cambridge, 2015.  doi: 10.1017/CBO9781316135914.  Google Scholar
[13]

T. Ilyashenko, Thick Attractors of step skew products, Regular and Chaotic Dynamics, 15 (2010), 328-334.  doi: 10.1134/S1560354710020188.  Google Scholar

[14]

A. Käenmäki, T. Rajala and V. Suomala, Local homogeneity and dimensions of measures in doubling metric spaces, Sc. Norm. Super. Pisa Cl. Sci., 16 (2016), 1315–1351, arXiv: 1003.2895v1.  Google Scholar

[15]

Y. Kifer, Thermodynamic formalism for random transformations revisited, Stoch. Dyn., 8 (2008), 77-102.  doi: 10.1142/S0219493708002238.  Google Scholar

[16]

R. Mañe, Ergodic Theory and Differentiable Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 8. Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-70335-5.  Google Scholar

[17]

J. N. Sarkooh, F. H. Ghane and A. Fakhari, Ergodic Properties and Thermodynamic Formalism of Markov Maps Induced by Locally Expanding Actions, arXiv: 1810.02299v2. Google Scholar

[18]

F. Oliveira and L. F. C. da Rocha, Minimal non-ergodic $C^1$-diffeomorphisms of the circle, Ergodic Theory Dynam. Systems, 21 (2001), 1843-1854.  doi: 10.1017/S0143385701001882.  Google Scholar

[19]

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

[20]

A. N. Quas, Non-ergodicity for $C^1$ expanding maps and $g$-measures, Ergodic Theory Dynam. Systems, 16 (1996), 531-543.  doi: 10.1017/S0143385700008956.  Google Scholar

[21]

A. A. Rashid and A. Zamani Bahabadi, Ergodicity of non-uniformly expanding transitive group (or semigroup) actions, J. Math. Phys., 57 (2016), 052702, 9 pp. doi: 10.1063/1.4947530.  Google Scholar

[22]

F. B. Rodrigues and P. Varandas, Specification and thermodynamical properties of semigroup actions, J. Math. Phys., 57 (2016), 052704, 27 pp. doi: 10.1063/1.4950928.  Google Scholar

[23]

M. Shub and D. Sullivan, Expanding endomorphisms of the circle revisited, Ergodic Theory Dynam. Systems, 5 (1985), 285-289.  doi: 10.1017/S014338570000290X.  Google Scholar

[24]

P. Varandas and M. Viana, Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 555-593.  doi: 10.1016/j.anihpc.2009.10.002.  Google Scholar

show all references

References:
[1]

J. F. Alves, SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Scient. Éc. Norm. Sup., 33 (2000), 1-32.  doi: 10.1016/S0012-9593(00)00101-4.  Google Scholar

[2]

J. F. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398.  doi: 10.1007/s002220000057.  Google Scholar

[3]

J. F. Alves, S. Luzzatto and V. Pinheiro, Markov structures and decay of correlations for non-uniformly expanding dynamical systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 817-839.  doi: 10.1016/j.anihpc.2004.12.002.  Google Scholar

[4]

J. F. Alves and H. Vilarinho, Strong stochastic stability for non-uniformly expanding maps, Ergod. Th. Dynam. Sys., 33 (2013), 647-692.  doi: 10.1017/S0143385712000077.  Google Scholar

[5]

P. G. Barrientos, A. Fakhari, D. Malicet and A. Sarizadeh, Expanding actions: Minimality and ergodicity, Stoch. Dyn., 17 (2017), 1750031, 20 pp. doi: 10.1142/S0219493717500319.  Google Scholar

[6]

R. Bowen, A horseshoe with positive measure, Invent. Math., 29 (1975), 203-204.  doi: 10.1007/BF01389849.  Google Scholar

[7]

Y. L. Cao, S. Luzzatto and I. Rios, Uniform hyperbolicity for random maps with positive Lyapunov exponents, Proc. Amer. Math. Soc., 136 (2008), 3591-3600.  doi: 10.1090/S0002-9939-08-09347-7.  Google Scholar

[8]

M. Denker and M. Urbański, On the existence of conformal measure, Trans. Amer. Math. Soc., 328 (1991), 563-587.  doi: 10.1090/S0002-9947-1991-1014246-4.  Google Scholar

[9]

E. Durand-Cartagena and J. A. Jaramillo, Pointwise Lipschitz functions on metric spaces, J. Math. Anal. Appl., 363 (2010), 525-548.  doi: 10.1016/j.jmaa.2009.09.039.  Google Scholar

[10]

A. Ehsani, F.-H. Ghane and M. Zaj, On Ergodicity of Mostly Expanding Semi-Group Actions, 2015, arXiv: 1505.03367. Google Scholar

[11]

M. Gharaei and A. J. Homburg, Random interval diffeomorphisms, Discrete & Continuous Dynamical Systems-S, 10 (2017), 241-272.  doi: 10.3934/dcdss.2017012.  Google Scholar

[12] J. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson, Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients, New Mathematical Monographs, 27. Cambridge University Press, Cambridge, 2015.  doi: 10.1017/CBO9781316135914.  Google Scholar
[13]

T. Ilyashenko, Thick Attractors of step skew products, Regular and Chaotic Dynamics, 15 (2010), 328-334.  doi: 10.1134/S1560354710020188.  Google Scholar

[14]

A. Käenmäki, T. Rajala and V. Suomala, Local homogeneity and dimensions of measures in doubling metric spaces, Sc. Norm. Super. Pisa Cl. Sci., 16 (2016), 1315–1351, arXiv: 1003.2895v1.  Google Scholar

[15]

Y. Kifer, Thermodynamic formalism for random transformations revisited, Stoch. Dyn., 8 (2008), 77-102.  doi: 10.1142/S0219493708002238.  Google Scholar

[16]

R. Mañe, Ergodic Theory and Differentiable Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 8. Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-70335-5.  Google Scholar

[17]

J. N. Sarkooh, F. H. Ghane and A. Fakhari, Ergodic Properties and Thermodynamic Formalism of Markov Maps Induced by Locally Expanding Actions, arXiv: 1810.02299v2. Google Scholar

[18]

F. Oliveira and L. F. C. da Rocha, Minimal non-ergodic $C^1$-diffeomorphisms of the circle, Ergodic Theory Dynam. Systems, 21 (2001), 1843-1854.  doi: 10.1017/S0143385701001882.  Google Scholar

[19]

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

[20]

A. N. Quas, Non-ergodicity for $C^1$ expanding maps and $g$-measures, Ergodic Theory Dynam. Systems, 16 (1996), 531-543.  doi: 10.1017/S0143385700008956.  Google Scholar

[21]

A. A. Rashid and A. Zamani Bahabadi, Ergodicity of non-uniformly expanding transitive group (or semigroup) actions, J. Math. Phys., 57 (2016), 052702, 9 pp. doi: 10.1063/1.4947530.  Google Scholar

[22]

F. B. Rodrigues and P. Varandas, Specification and thermodynamical properties of semigroup actions, J. Math. Phys., 57 (2016), 052704, 27 pp. doi: 10.1063/1.4950928.  Google Scholar

[23]

M. Shub and D. Sullivan, Expanding endomorphisms of the circle revisited, Ergodic Theory Dynam. Systems, 5 (1985), 285-289.  doi: 10.1017/S014338570000290X.  Google Scholar

[24]

P. Varandas and M. Viana, Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 555-593.  doi: 10.1016/j.anihpc.2009.10.002.  Google Scholar

Figure 1.  Diffeomorphisms $ f_0 $, $ f_1 $
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