Thermodynamic formalism for Lorenz maps

TL;DR

This paper develops a thermodynamic formalism for Lorenz maps, characterizing attractors and equilibrium states, and establishing conditions for their existence and uniqueness.

Contribution

It introduces the concept of good measures for Lorenz attractors and proves uniqueness of equilibrium states among these measures for H"older potentials.

Findings

Attractors are closures of unions of unstable leaves.

Existence conditions for equilibrium states are provided.

Uniqueness of equilibrium states among good measures is established.

Abstract

For a 2-dimensional map representing an expanding geometric Lorenz at- tractor we prove that the attractor is the closure of a union of as long as possible unstable leaves with ending points. This allows to define the notion of good measures, those giving full measure to the union of these open leaves. Then, for any H\"older continuous potential we prove that there exists at most one relative equilibrium state among the set of good measures. Condition yielding existence are given.