Phase Transitions for one-dimensional Lorenz-like expanding Maps

Abstract

Given an one-dimensional Lorenz-like expanding map we prove that the condition\linebreak $P_{top}(\varphi ,\partial \mathcal{P},\ell )<P_{top}(\varphi ,\ell )$ (see, subsection 2.4 for definition), introduced by Buzzi and Sarig in [1] is satisfied for all continuous potentials $\varphi :[0,1]⟶\mathbb{R}$. We apply this to prove that quasi-Hölder-continuous potentials (see, subsection 2.2 for definition) have at most one equilibrium measure and we construct a family of continuous but not Hölder and neither weak Hölder continuous potentials for which we observe phase transitions. Indeed, this class includes all Hölder and weak-Hölder continuous potentials and form an open and [2].