Phase Transitions for One-Dimensional Lorenz-Like Expanding Maps

Abstract

Given an one-dimensional Lorenz-like expanding map we describe a class A of potentials ϕ: [0 , 1] ⟶ R admitting at most one equilibrium measure and we construct a family of continuous but not weak-Hölder continuous potentials for which we observe phase transitions. This give a certain generalization of the results proved in Pesin and Zhang (J Stat Phys 122(6):1095–1110, 2006), where the authors have proved this for a smaller class of potentials, that is, for uniformly expanding maps and weak-Hölder continuous potentials. Indeed, the class A form an open and dense subset of C([0 , 1] , R) , with the usual C topology.