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

Published online by Cambridge University Press:  12 May 2016

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 paulo.varandas@ufba.br
YUN ZHAO
Affiliation:
Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil

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
Copyright
© Cambridge University Press, 2016 

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