A uniform estimate for rate functions in large deviations

Abstract

Given Holder continuous functions f and \(\psi \) on a subshift of finite type \(\mathrm{\Sigma }_{A}^{+}\) such that \(\psi \) is not cohomologous to a constant, the classical large deviation principle holds with a rate function \(I_\psi \geqslant 0\) such that \(I_\psi (p) = 0\) iff Open image in new window, where Open image in new window is the equilibrium state of f. In this paper we derive a uniform estimate from below for \(I_\psi \) for p outside an interval containing Open image in new window, which depends only on the subshift \(\mathrm{\Sigma }_{A}^{+}\), the function f, the norm \(|\psi |_\infty \), the Holder constant of \(\psi \) and the integral \(\widetilde{\psi }\). Similar results can be derived in the same way, e.g. for Axiom A diffeomorphisms on basic sets.