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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
| 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 |
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.
Mathematics Subject Classification:
37C40, 37C60.
Citation:
Pablo G. Barrientos, Abbas Fakhari. Ergodicity of non-autonomous discrete systems with non-uniform expansion. Discrete and Continuous Dynamical Systems - B,
2020, 25
(4)
: 1361-1382.
doi: 10.3934/dcdsb.2019231
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [6] |
R. Bowen,
A horseshoe with positive measure, Invent. Math., 29 (1975), 203-204.
doi: 10.1007/BF01389849. |
| [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. |
| [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. |
| [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. |
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A. Ehsani, F.-H. Ghane and M. Zaj, On Ergodicity of Mostly Expanding Semi-Group Actions, 2015, arXiv: 1505.03367. |
| [11] |
M. Gharaei and A. J. Homburg,
Random interval diffeomorphisms, Discrete & Continuous Dynamical Systems-S, 10 (2017), 241-272.
doi: 10.3934/dcdss.2017012. |
| [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.
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T. Ilyashenko,
Thick Attractors of step skew products, Regular and Chaotic Dynamics, 15 (2010), 328-334.
doi: 10.1134/S1560354710020188. |
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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. |
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Y. Kifer,
Thermodynamic formalism for random transformations revisited, Stoch. Dyn., 8 (2008), 77-102.
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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. |
| [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. |
| [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.
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| [19] |
V. Pinheiro,
Expanding measures, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 889-939.
doi: 10.1016/j.anihpc.2011.07.001. |
| [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. |
| [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. |
| [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. |
| [23] |
M. Shub and D. Sullivan,
Expanding endomorphisms of the circle revisited, Ergodic Theory Dynam. Systems, 5 (1985), 285-289.
doi: 10.1017/S014338570000290X. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [6] |
R. Bowen,
A horseshoe with positive measure, Invent. Math., 29 (1975), 203-204.
doi: 10.1007/BF01389849. |
| [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. |
| [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. |
| [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. |
| [10] |
A. Ehsani, F.-H. Ghane and M. Zaj, On Ergodicity of Mostly Expanding Semi-Group Actions, 2015, arXiv: 1505.03367. |
| [11] |
M. Gharaei and A. J. Homburg,
Random interval diffeomorphisms, Discrete & Continuous Dynamical Systems-S, 10 (2017), 241-272.
doi: 10.3934/dcdss.2017012. |
| [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.
|
| [13] |
T. Ilyashenko,
Thick Attractors of step skew products, Regular and Chaotic Dynamics, 15 (2010), 328-334.
doi: 10.1134/S1560354710020188. |
| [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. |
| [15] |
Y. Kifer,
Thermodynamic formalism for random transformations revisited, Stoch. Dyn., 8 (2008), 77-102.
doi: 10.1142/S0219493708002238. |
| [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. |
| [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. |
| [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. |
| [19] |
V. Pinheiro,
Expanding measures, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 889-939.
doi: 10.1016/j.anihpc.2011.07.001. |
| [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. |
| [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. |
| [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. |
| [23] |
M. Shub and D. Sullivan,
Expanding endomorphisms of the circle revisited, Ergodic Theory Dynam. Systems, 5 (1985), 285-289.
doi: 10.1017/S014338570000290X. |
| [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. |
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