Zero-temperature limit of one-dimensional Gibbs states via renormalization: the case of locally constant potentials

Let $A$ be a finite set and $\varphi :A^Z\to R$ be a locally constant potential. For each $\beta >0$ (“inverse temperature”), there is a unique Gibbs measure ${\mu }_{\beta \varphi }$. We prove that, as $\beta \to +\mathrm{\infty }$, the family $({\mu }_{\beta \varphi }{)}_{\beta >0}$ converges (in weak-${}^{\ast }$ topology) to a measure we characterize. It is concentrated on a certain subshift of finite type which is a finite union of transitive subshifts of finite type. The two main tools are an approximation by periodic orbits and the Perron-Frobenius Theorem for matrices 'a la Birkhoff. The crucial idea we bring is a “renormalization” procedure which explains convergence and provides a recursive algorithm to compute the weights of the ergodic decomposition of the limit.