Equilibrium measure for one-dimensional Lorenz-like expanding Maps

Let $L:[0,1]\setminus \{d\}\to [0,1]$ be a one-dimensional Lorenz like expanding map ($d$ is the point of discontinuity), $$\mathcal{P}$={ (0,d),(d,1) }$beapartitionof$[0,1]$and$C^{\alpha}([0,1],$\mathcal{P}$)$ the set of piecewise H"older-continuous potential of [0,1] with the usual ${\mathcal{C}}^0$ topology. In this context, we prove, improving a result of \cite{BS03}, that piecewise H"older-continuous potential $\varphi$ satisfying \linebreak $\max\left{ \limsup_{n \rightarrow \infty}\frac{1}{n}(S_{n}\phi)(0),\limsup_{n \rightarrow \infty}\frac{1}{n}(S_{n}\phi)(1)\right}<P_{$\text{top}$}(\phi,T)$ support an unique equilibrium state. Indeed, we prove there exists an open and dense subset $\mathcal{H}$ of $C^{\alpha }([0,1],\mathcal{P})$ such that, if $\varphi \in \mathcal{H}$, then $\varphi$ admits one equilibrium measure.