Phase Transitions for one-dimensional Lorenz-like expanding Maps

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