Let $\Sigma_A$ be a topologically mixing subshift of finite type, and let $\phi:\Sigma_A\to\mathbb R$ be Hölder continuous. Then there exist $\lambda>0$, a strictly positive Hölder function $h$, and a Borel probability measure $\nu$ such that \begin{align*} \mathcal L_\phi h=\lambda h,\qquad \mathcal L_\phi^*\nu=\lambda \nu,\qquad \int h\,d\nu=1. \end{align*} Moreover $\lambda=e^{P(\sigma,\phi)}$, the measure $\mu=h\nu$ is $\sigma$-invariant, and for every Hölder function $f$, \begin{align*} \lambda^{-n}\mathcal L_\phi^n f\to h\int f\,d\nu \end{align*} uniformly.