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 Iψ  0 such that Iψ(p) = 0 iff p =  ψdμ, where μ = μf
is the equilibrium state of f . In this paper we derive a uniform estimate from below
for Iψ for p outside an interval containing ψ  =  ψdμ, 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.

• A uniform estimate for rate functions in large deviations