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

Abstract

Abstract 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 μ β ϕ . We prove that as β →+ ∞ , the family ( μ β ϕ ) 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.