A uniform estimate for rate functions in large deviations

Abstract

Given Hölder continuous functions f and ψ on a subshift of finite type + A such that ψ is not cohomologous to a constant, the classical large deviation principle holds with a rate function I ψ 0 such that I ψ ( p) = 0 iff p = ψ dμ, where μ = μ f is the equilibrium state of f . In this paper we derive a uniform estimate from below for I ψ for p outside an interval containing ψ = ψ dμ, which depends only on the subshift + A , the function f , the norm |ψ| ∞ , the Hölder constant of ψ and the integral ψ. Similar results can be derived in the same way, e.g. for Axiom A diffeomorphisms on basic sets.