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

Let $A$ be a finite set and $ϕ:A^Z\to R$ be a locally constant potential. For each $β>0$ ("inverse temperature"), there is a unique Gibbs measure $μ_{βϕ}$. We prove that, as $β\to+\infty$, the family $(μ_{βϕ})_{β>0}$ converges (in weak-$^*$ 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 á 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.