Large deviations estimates for dynamical systems without the specification property. Application to the β-shifts

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Published 8 October 2004 2005 IOP Publishing Ltd and London Mathematical Society
, , Citation C-E Pfister and W G Sullivan 2005 Nonlinearity 18 237 DOI 10.1088/0951-7715/18/1/013

0951-7715/18/1/237

Abstract

We consider dynamical systems whose sets of orbits verify an approximate product property. This allows us to obtain large deviations results, which were previously proven under the condition of specification property. We illustrate our results by considering the β-shifts. While the specification property holds for a set of β > 1 of Lebesgue measure zero, our approximate product property holds for any β > 1. For any β-shift the empirical measures verify a large deviations principle with respect to the probability measure of maximal entropy. We extend the dimension results of Pfister and Sullivan (2003 Nonlinearity 16 661–82) to any β-shift, obtaining a variational principle for the topological entropy of sets involving ergodic averages.

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