Let $\Sigma_A$ be a one-sided topologically mixing subshift of finite type over a finite alphabet, let $\sigma: \Sigma_A \to \Sigma_A$ be the left shift, and let $\phi: \Sigma_A \to \mathbb{R}$ be Hölder continuous. Suppose the [Ruelle-Perron-Frobenius theorem](/theorems/6817) supplies $\lambda > 0$, a strictly positive continuous eigenfunction $h: \Sigma_A \to (0,\infty)$, and a Borel probability measure $\nu$ on $\Sigma_A$ such that

\begin{align*} \mathcal{L}_\phi h = \lambda h, \end{align*}

\begin{align*} \mathcal{L}_\phi^*\nu = \lambda \nu, \end{align*}

and

\begin{align*} \int_{\Sigma_A} h \, d\nu(y) = 1. \end{align*}

Define the invariant probability measure $\mu$ on $\Sigma_A$ by

\begin{align*} \mu(B) = \int_B h \, d\nu(y) \end{align*}

for every Borel set $B \subset \Sigma_A$.

Then $\mu$ is a Gibbs measure for $\phi$: there exists a constant $C \geq 1$ such that, for every admissible word $w = w_0\cdots w_{n-1}$ of length $n \geq 1$ and every point $x \in [w]$,

\begin{align*} C^{-1} \leq \frac{\mu([w])}{\exp(S_n\phi(x) - nP)} \leq C, \end{align*}

where

\begin{align*} S_n\phi(x) = \sum_{k=0}^{n-1}\phi(\sigma^k x) \end{align*}

and $P = \log \lambda$.