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

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 beta-shifts. While the specification property holds for a set of beta > 1 of Lebesgue measure zero, our approximate product property holds for any beta > 1. For any beta-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 beta-shift, obtaining a variational principle for the topological entropy of sets involving ergodic averages.