Equilibrium measure for one-dimensional Lorenz-like expanding Maps

Abstract

Let $L:[0,1]\setminus \{d\}\to [0,1]$ be a one-dimensional Lorenz like expanding map ($d$ is the point of discontinuity), $\mathcal{P}=\{(0,d),(d,1)\}$ be a partition of $[0,1]$ and $C^{\alpha }([0,1],\mathcal{P})$ the set of piecewise Hölder-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 Hölder-continuous potential $\varphi$ satisfying \linebreak $\max \left\{{\mathrm{limsup}}_{n\to \infty }\frac{1}{n}(S_n\varphi )(0),{\mathrm{limsup}}_{n\to \infty }\frac{1}{n}(S_n\varphi )(1)\right\}<P_{\text{top}}(\varphi ,T)$ 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.

18 pages, 3 figures