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

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