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

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

Ground states are generically a periodic orbit

“…Brémont [5] proves that the limit lim β→+∞ µ βF exists if F is locally constant. Chazottes, Gambaudo and Ugalde [8] give a characterization of the limit and a new proof of Brémont's Gonzalo Contreras was Partially supported by conacyt, Mexico, grant 178838.…”

On the computability of rotation sets and their entropies

“…Next, we prove (ii): Let µ n be as in Equation (15). We claim that rv Φε n (µ n+1 ) ∈ B(w 0 , αε n ).…”

Ergodic optimization in dynamical systems

“…For larger p, and for other subshifts of finite type (X, T ), the set of possible limits L p (X) becomes harder to describe. Progress on this problem was made initially by Leplaideur [107], then by Chazottes, Gambaudo & Ugalde [45] and Garibaldi & Thieullen [67], using a variety of techniques, and can be summarised as follows: Theorem 4.3. (Description of zero temperature limit for locally constant functions) If (X, T ) is a subshift of finite type, and f : X → R is locally constant, then m = lim t→∞ m tf is concentrated on a certain subshift of finite type X f which is itself a finite union of transitive subshifts of finite type.…”