Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T05:17:59.276Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak specification properties and large deviations for non-additive potentials

Published online by Cambridge University Press:  09 October 2013

PAULO VARANDAS  and
YUN ZHAO
PAULO VARANDAS
Affiliation:
Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil email [email protected]
YUN ZHAO
Affiliation:
Departament of Mathematics, Soochow University, Suzhou 215006, Jiangsu, P.R. China email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We obtain large deviation bounds for the measure of deviation sets associated with asymptotically additive and sub-additive potentials under some weak specification properties. In particular, a large deviation principle is obtained in the case of uniformly hyperbolic dynamical systems. Some applications to the study of the convergence of Lyapunov exponents are given.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 3 , May 2015 , pp. 968 - 993
Copyright
© Cambridge University Press, 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Araújo, V. and Pacífico, M. J.. Large deviations for non-uniformly expanding maps. J. Stat. Phys. 125 (2006), 415–457.CrossRefGoogle Scholar
Ban, J., Cao, Y. and Hu, H.. The dimensions of non-conformal repeller and average conformal repeller. Trans. Amer. Math. Soc. 362 (2010), 727–751.CrossRefGoogle Scholar
Barreira, L.. A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems. Ergod. Th. & Dynam. Sys. 16 (1996), 871–927.Google Scholar
Barreira, L.. Nonadditive thermodynamic formalism: equilibrium and Gibbs measures. Discrete Contin. Dyn. Syst. 16 (2006), 279–305.Google Scholar
Barreira, L. and Gelfert, K.. Multifractal analysis for Lyapunov exponents on nonconformal repellers. Comm. Math. Phys. 267 (2006), 393–418.Google Scholar
Blokh, A. M.. Decomposition of dynamical systems on an interval. Uspekhi Mat. Nauk 38 (1983), 179–180.Google Scholar
Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401–414.Google Scholar
Bowen, R.. Some systems with unique equilibrium states. Math. Systems Theory 8 (1974), 193–202.Google Scholar
Brin, M. and Katok, A.. On local entropy. Geometric Dynamics (Rio de Janeiro) (Lecture Notes in Mathematics, 1007). Springer, New York, 1983.Google Scholar
Buzzi, J.. Specification on the interval. Trans. Amer. Math. Soc. 349 (1997), 2737–2754.CrossRefGoogle Scholar
Chung, Y. M.. Large deviations on Markov towers. Nonlinearity 24 (2011), 1229–1252.CrossRefGoogle Scholar
Cao, Y., Feng, D. and Huang, W.. The thermodynamic formalism for sub-additive potentials. Discrete Contin. Dyn. Syst. 20 (2008), 639–657.Google Scholar
Cao, Y., Hu, H. and Zhao, Y.. Nonadditive measure-theoretic pressure and applications to dimensions of an ergodic measure. Ergod. Th. & Dynam. Sys. 33 (2013), 831–850.Google Scholar
Cheng, W., Zhao, Y. and Cao, Y.. Pressures for asymptotically subadditive potentials under a mistake funciton. Discrete Contin. Dyn. Syst. 32 (2) (2012), 487–497.Google Scholar
Climenhaga, V., Thompson, D. and Yamamoto, K.. Large deviations for systems with non-uniform structure. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 539–579.Google Scholar
Comman, H. and Rivera-Letelier, J.. Large deviations principles for non-uniformly hyperbolic rational maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 539–579.Google Scholar
Comman, H.. Strengthened large deviations for rational maps and full shifts, with unified proof. Nonlinearity 22 (2009), 1413–1429.Google Scholar
Eizenberg, A., Kifer, Y. and Weiss, B.. Large deviations for ${ \mathbb{Z} }^{d} $-actions. Commun. Math. Phys. 164 (1994), 433–454.Google Scholar
Feng, D.-J. and Lau, K.-S.. The pressure function for products of non-negative matrices. Math. Res. Lett. 9 (2002), 363–378.Google Scholar
Feng, D. and Käenmäki, A.. Equilibrium states of the pressure function for products of matrices. Discrete Contin. Dyn. Syst. 30 (3) (2011), 699–708.CrossRefGoogle Scholar
Feng, D. and Huang, W.. Lyapunov spectrum of asymptotically sub-additive potentials. Commun. Math. Phys. 297 (2010), 1–43.Google Scholar
Iommi, G. and Yayama, Y.. Almost-additive thermodynamical formalism for countable Markov shifts. Nonlinearity 25 (2012), 165–191.Google Scholar
Katok, A.. Lyapunov exponents, entropy and periodic points of diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173.Google Scholar
Kifer, Y.. Large deviations in dynamical systems and stochastic processes. Trans. Amer. Math. Soc. 321 (2) (1990), 505–524.Google Scholar
Kifer, Y. and Newhouse, S. E.. A global volume lemma and applications. Israel J. Math. 74 (2–3) (1991), 209–223.Google Scholar
Melbourne, I.. Large and moderate deviations for slowly mixing dynamical systems. Proc. Amer. Math. Soc. 137 (2009), 1735–1741.Google Scholar
Melbourne, I. and Nicol, M.. Large deviations for non-uniformly hyperbolic systems. Trans. Amer. Math. Soc. 360 (2008), 6661–6676.CrossRefGoogle Scholar
Méson, A. and Vericat, F.. Estimates of large deviations in dynamical systems by a non-additive thermodynamic formalism. Far East J. Dyn. Syst. 11 (2009), 1–16.Google Scholar
Mummert, A.. The thermodynamic formalism for almost-additive sequences. Discrete Contin. Dyn. Syst. 16 (2006), 435–454.CrossRefGoogle Scholar
Pfister, C.-E. and Sullivan, W. G.. Billingsley dimension on shift spaces. Nonlinearity 16 (2003), 661–682.Google Scholar
Pfister, C.-E. and Sullivan, W.. Large deviations estimates for dynamical systems without the specification property. Application to the beta-shifts. Nonlinearity 18 (2005), 237–261.Google Scholar
Pfister, C.-E. and Sullivan, W. G.. On the topological entropy of saturated sets. Ergod. Th. & Dynam. Sys. 27 (2007), 929–956.CrossRefGoogle Scholar
Pollicott, M. and Sharp, R.. Large deviations for intermittent maps. Nonlinearity 22 (2009), 2079–2092.Google Scholar
Rényi, A.. Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493.Google Scholar
Rey-Bellet, L. and Young, L.-S.. Large deviations in non-uniformly hyperbolic dynamical systems. Ergod. Th. & Dynam. Sys. 28 (2) (2008), 587–612.CrossRefGoogle Scholar
Rousseau, J., Varandas, P. and Zhao, Y.. Entropy formula for dynamical systems with mistakes. Discrete Contin. Dyn. Syst. 32 (12) (2012), 4391–4407.Google Scholar
Saussol, B., Troubetzkoy, S. and Vaienti, S.. Recurrence and Lyapunov exponents. Mosc. Math. J. 3 (2003), 189–203.Google Scholar
Sakai, K., Sumi, N. and Yamamoto, K.. Diffeomorphisms satisfying the specification property. Proc. Amer. Math. Soc. 138 (2009), 315–321.Google Scholar
Sumi, N., Varandas, P. and Yamamoto, K.. Partial hyperbolicity and specification. Preprint, 2013.Google Scholar
Thompson, D.. Irregular sets, the beta-transformation and the almost specification property. Trans. Amer. Math. Soc. 364 (2012), 5395–5414.Google Scholar
Varandas, P. and Viana, M.. Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps. Ann. Inst. H. Poincaré Anal. Non-Lineaire 27 (2010), 555–593.CrossRefGoogle Scholar
Varandas, P.. Non-uniform specification and large deviations for weak Gibbs measures. J. Statist. Phys. 146 (2012), 330–358.CrossRefGoogle Scholar
Young, L.-S.. Some large deviations for dynamical systems. Trans. Amer. Math. Soc. 318 (1990), 525–543.Google Scholar
Yuri, M.. Weak Gibbs measures for certain non-hyperbolic systems. Ergod. Th. & Dynam. Sys. 20 (2000), 1495–1518.Google Scholar
Yuri, M.. Large deviations for countable to one Markov systems. Comm. Math. Phys. 258 (2005), 455–474.Google Scholar
Zhao, Y., Zhang, L. and Cao, Y.. The asymptotically additive topological pressure on the irregular set for asymptotically additive potentials. Nonlinear Anal. 74 (2011), 5015–5022.Google Scholar