Large Deviations Estimates for Dynamical Systems without the Specification Property. Application to the β-shifts, Nonlinearity 18

We consider dynamical systems whose sets of orbits verify an approximate prod-uct property. This allows us to obtain large deviations results, which were previ-ously 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 re-spect to the probability measure of maximal entropy. We extend the dimension results of [PfS1] to any β-shift, obtaining a variational principle for the topological entropy of sets involving ergodic averages.