Equilibrium measure for one-dimensional Lorenz-like expanding Maps

TL;DR

This paper establishes the existence and uniqueness of equilibrium measures for a class of one-dimensional Lorenz-like expanding maps with piecewise Hölder potentials, extending previous results by identifying an open dense subset of potentials with unique equilibrium states.

Contribution

It improves prior work by proving that a broad class of piecewise Hölder potentials on Lorenz-like maps have unique equilibrium measures, characterizing conditions for their existence.

Findings

Unique equilibrium measure exists for generic potentials.

Open and dense subset of potentials admits a unique equilibrium state.

Conditions on potentials ensure support of the equilibrium measure.

Abstract

Let $L:[0,1]\setminus\{d\}\rightarrow [0,1]$ 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 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 $\phi$ 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 $\phi \in \mathcal{H}$, then $\phi$ admits one equilibrium measure.