Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T00:37:22.740Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak Gibbs measures: speed of convergence to entropy, topological and geometrical aspects

Published online by Cambridge University Press:  12 May 2016

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 CMUP, University of Porto, Portugal email [email protected]
YUN ZHAO
Affiliation:
Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we obtain exponential large-deviation bounds in the Shannon–McMillan–Breiman convergence formula for entropy in the case of weak Gibbs measures and topologically mixing subshifts of finite type. We also prove almost sure estimates for the error term in the convergence to entropy given by the Shannon–McMillan–Breiman formula for both uniformly and non-uniformly expanding shifts. Finally, we establish a topological characterization of large-deviation bounds for Gibbs measures and deduce some of their topological and geometrical aspects: the local entropy is zero and the topological pressure of positive measure sets is total. Some applications include large-deviation estimates for Lyapunov exponents, pointwise dimension and slow return times.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 7 , October 2017 , pp. 2313 - 2336
Copyright
© Cambridge University Press, 2016 

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

Barreira, L.. Nonadditive thermodynamic formalism: equilibrium and Gibbs measures. Discrete Contin. Dyn. Syst. 16 (2006), 279–305.CrossRefGoogle Scholar
Bomfim, T. and Varandas, P.. Multifractal analysis of the irregular set for almost-additive sequences via large deviations. Nonlinearity 28 (2015), 3563–3585.CrossRefGoogle Scholar
Bomfim, T. and Varandas, P.. Multifractal analysis for weak Gibbs measures: from large deviations to irregular sets. Ergod. Th. & Dynam. Sys., 2015, to appear, doi:10.1017/etds.2015.46.Google Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470) . Springer, Berlin, 1975.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
Downarowicz, T.. Entropy in Dynamical Systems (New Mathematical Monographs, 18) . Cambridge University Press, Cambridge, 2011.Google Scholar
Feng, D. and Huang, W.. Lyapunov spectrum of asymptotically sub-additive potentials. Comm. Math. Phys. 297 (2010), 1–43.Google Scholar
Haydn, N. and Vaienti, S.. The Rényi entropy function and the large deviation of short return times. Ergod. Th. & Dynam. Sys. 30(1) (2010), 159–179.Google Scholar
Iommi, G. and Yayama, Y.. Weak Gibbs measures as Gibbs measures for aysmptotically additive sequences. Preprint, 2015, arXiv:1505.00977.CrossRefGoogle Scholar
Jordan, T. and Rams, M.. Multifractal analysis of weak Gibbs measures for non-uniformly expanding C 1 maps. Ergod. Th. & Dynam. Sys. 31(1) (2011), 143–164.Google Scholar
Katok, A.. Fifty years of entropy in dynamics: 1958–2007. J. Mod. Dyn. 1 (2007), 545–596.CrossRefGoogle Scholar
Katznelson, Y. and Weiss, B.. A simple proof of some ergodic theorems. Israel J. Math. 42(4) (1982), 291–296.Google Scholar
Kleptsyn, V., Ryzhov, D. and Minkov, S.. Special ergodic theorems and dynamical large deviations. Nonlinearity 25 (2012), 3189–3196.Google Scholar
Melbourne, I. and Nicol, M.. Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys. 260(1) (2005), 131–146.Google Scholar
Mummert, A.. The thermodynamic formalism for almost-additive sequences. Discrete Contin. Dyn. Syst. 16 (2006), 435–454.CrossRefGoogle Scholar
Philipp, W. and Stout, W.. Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. 2(161) (1975).Google 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
Varandas, P.. Correlation decay and recurrence asymptotics for some robust nonuniformly hyperbolic maps. J. Stat. Phys. 133(5) (2008), 813–839.CrossRefGoogle Scholar
Varandas, P.. Entropy and Poincaré recurrence from a geometrical viewpoint. Nonlinearity 22 (2009), 2365–2375.Google Scholar
Varandas, P. and Zhao, Y.. Weak specification properties and large deviations for non-additive potentials. Ergod. Th. & Dynam. Sys. 35(3) (2015), 968–993.Google Scholar
Walters, P.. An Introduction to Ergodic Theory. Springer, New York, 1982.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