American Institute of Mathematical Sciences

January  2016, 36(1): 1-41. doi: 10.3934/dcds.2016.36.1

Statistical properties of diffeomorphisms with weak invariant manifolds

 1 Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal, Portugal

Received  August 2014 Revised  March 2015 Published  June 2015

We consider diffeomorphisms of compact Riemmanian manifolds which have a Gibbs-Markov-Young structure, consisting of a reference set $\Lambda$ with a hyperbolic product structure and a countable Markov partition. We assume polynomial contraction on stable leaves, polynomial backward contraction on unstable leaves, a bounded distortion property and a certain regularity of the stable foliation. We establish a control on the decay of correlations and large deviations of the physical measure of the dynamical system, based on a polynomial control on the Lebesgue measure of the tail of return times. Finally, we present an example of a dynamical system defined on the torus and prove that it verifies the properties of the Gibbs-Markov-Young structure that we considered.
Citation: José F. Alves, Davide Azevedo. Statistical properties of diffeomorphisms with weak invariant manifolds. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 1-41. doi: 10.3934/dcds.2016.36.1
References:
 [1] J. F. Alves and V. Pinheiro, Slow rates of mixing for dynamical systems with hyperbolic structures, J. Stat. Phys., 131 (2008), 505-534. doi: 10.1007/s10955-008-9482-6. [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] V. Araújo, Large deviations for semiflows over a non-uniformly expanding base, Bull. Braz. Math. Soc. (N.S.), 38 (2007), 335-376. doi: 10.1007/s00574-007-0049-y. [4] V. Araújo and M. J. Pacifico, Large deviations for non-uniformly expanding maps, J. Stat. Phys., 125 (2006), 415-457. doi: 10.1007/s10955-006-9183-y. [5] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, vol. 470, Springer, New York, 1975. [6] M. Benedicks and L.-S. Young, Markov extensions and decay of correlations for certain Hénon maps, Astérisque, 261 (2000), 13-56. [7] Y. Kifer, Large deviations in dynamical systems and stochastic processes, Trans. Amer. Math. Soc., 321 (1990), 505-524. doi: 10.1090/S0002-9947-1990-1025756-7. [8] A. Lopes, Entropy and large deviations, Nonlinearity, 3 (1990), 527-546. doi: 10.1088/0951-7715/3/2/013. [9] R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-70335-5. [10] I. Melbourne, Large and moderate deviations for slowly mixing dynamical systems, Proc. Amer. Math. Soc., 137 (2009), 1735-1741. doi: 10.1090/S0002-9939-08-09751-7. [11] I. Melbourne and M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Commun. Math. Phys., 260 (2005), 131-146. doi: 10.1007/s00220-005-1407-5. [12] I. Melbourne and M. Nicol, Large deviations for nonuniformly hyperbolic systems, Trans. Amer. Math. Soc., 360 (2000), 6661-6676. doi: 10.1090/S0002-9947-08-04520-0. [13] S. Orei and S. Pelikan, Large deviations principles for stationary processes, Ann. Probab., 16 (1988), 1481-1495. doi: 10.1214/aop/1176991579. [14] D. Ruelle, A measure associated with Axiom A attractors, Am. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810. [15] E. Rio, Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants, Mathématiques & Applications(Berlin) [Mathematics and Applications] 31, Springer-Verlag, Berlin, 2000. [16] Ya. Sinai, Gibbs measure in ergodic theory, Russ. Math. Surv., 27 (1972), 21-64. [17] S. Waddington, Large deviations asymptotics for Anosov flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 445-484. [18] L.-S. Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543. doi: 10.2307/2001318. [19] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147 (1998), 585-650. doi: 10.2307/120960. [20] 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] J. F. Alves and V. Pinheiro, Slow rates of mixing for dynamical systems with hyperbolic structures, J. Stat. Phys., 131 (2008), 505-534. doi: 10.1007/s10955-008-9482-6. [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] V. Araújo, Large deviations for semiflows over a non-uniformly expanding base, Bull. Braz. Math. Soc. (N.S.), 38 (2007), 335-376. doi: 10.1007/s00574-007-0049-y. [4] V. Araújo and M. J. Pacifico, Large deviations for non-uniformly expanding maps, J. Stat. Phys., 125 (2006), 415-457. doi: 10.1007/s10955-006-9183-y. [5] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, vol. 470, Springer, New York, 1975. [6] M. Benedicks and L.-S. Young, Markov extensions and decay of correlations for certain Hénon maps, Astérisque, 261 (2000), 13-56. [7] Y. Kifer, Large deviations in dynamical systems and stochastic processes, Trans. Amer. Math. Soc., 321 (1990), 505-524. doi: 10.1090/S0002-9947-1990-1025756-7. [8] A. Lopes, Entropy and large deviations, Nonlinearity, 3 (1990), 527-546. doi: 10.1088/0951-7715/3/2/013. [9] R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-70335-5. [10] I. Melbourne, Large and moderate deviations for slowly mixing dynamical systems, Proc. Amer. Math. Soc., 137 (2009), 1735-1741. doi: 10.1090/S0002-9939-08-09751-7. [11] I. Melbourne and M. Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Commun. Math. Phys., 260 (2005), 131-146. doi: 10.1007/s00220-005-1407-5. [12] I. Melbourne and M. Nicol, Large deviations for nonuniformly hyperbolic systems, Trans. Amer. Math. Soc., 360 (2000), 6661-6676. doi: 10.1090/S0002-9947-08-04520-0. [13] S. Orei and S. Pelikan, Large deviations principles for stationary processes, Ann. Probab., 16 (1988), 1481-1495. doi: 10.1214/aop/1176991579. [14] D. Ruelle, A measure associated with Axiom A attractors, Am. J. Math., 98 (1976), 619-654. doi: 10.2307/2373810. [15] E. Rio, Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants, Mathématiques & Applications(Berlin) [Mathematics and Applications] 31, Springer-Verlag, Berlin, 2000. [16] Ya. Sinai, Gibbs measure in ergodic theory, Russ. Math. Surv., 27 (1972), 21-64. [17] S. Waddington, Large deviations asymptotics for Anosov flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 445-484. [18] L.-S. Young, Large deviations in dynamical systems, Trans. Amer. Math. Soc., 318 (1990), 525-543. doi: 10.2307/2001318. [19] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147 (1998), 585-650. doi: 10.2307/120960. [20] L.-S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180.
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