A uniform estimate for rate functions in large deviations

Given H"older continuous functions $f$ and $\psi$ on a sub-shift of finite type ${\mathrm{\Sigma }}_A^{+}$ such that $\psi$ is not cohomologous to a constant, the classical large deviation principle holds (\cite{OP}, \cite{Kif}, \cite{Y}) with a rate function $I_{\psi }\ge 0$ such that $I_{\psi }(p)=0$ iff $p = \int \psi , d \mu$,where$\mu = \mu_f$istheequilibriumstateof$f$. In this paper we derive a uniform estimate from below for $I_{\psi }$ for $p$ outside an interval containing $\tilde{\psi }=\int \psi d\mu$, which depends only on the sub-shift, the function $f$, the norm $\mathrm{\mid }\psi {\mathrm{\mid }}_{\mathrm{\infty }}$, the H"older constant of $\psi$ and the integral $\tilde{\psi }$. Similar results can be derived in the same way e.g. for Axiom A diffeomorphisms on basic sets.