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

Let A be a finite set and let φ:A^ℤ→ℝ be a locally constant potential. For each β>0 ('inverse temperature'), there is a unique Gibbs measure μβφ. We prove that as β→+∞ , the family (μβφ)β>0 converges (in the weak-* topology) to a measure that we characterize. This measure 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 à la Birkhoff. The crucial idea we bring is a 'renormalization' procedure which explains convergence and provides a recursive algorithm for computing the weights of the ergodic decomposition of the limit.