Equilibrium states for the classical Lorenz attractor and sectional-hyperbolic attractors in higher dimensions

TL;DR

This paper proves the uniqueness of equilibrium states, including the measure of maximal entropy, for the classical Lorenz attractor and higher-dimensional sectional-hyperbolic attractors under certain conditions.

Contribution

It establishes the uniqueness of equilibrium states for a broad class of hyperbolic attractors, confirming long-standing conjectures in dynamical systems theory.

Findings

Unique measure of maximal entropy exists for the classical Lorenz attractor.

In a dense family of vector fields, equilibrium states are unique if point masses at singularities are not equilibrium states.

The results extend to higher-dimensional sectional-hyperbolic attractors.

Abstract

It has long been conjectured that the classical Lorenz attractor supports a unique measure of maximal entropy. In this article, we give a positive answer to this conjecture and its higher-dimensional counterpart by considering the uniqueness of equilibrium states for H\"older continuous functions on a sectional-hyperbolic attractor $\Lambda$. We prove that in a $C^1$-open and dense family of vector fields (including the classical Lorenz attractor), if the point masses at singularities are not equilibrium states, then there exists a unique equilibrium state supported on $\Lambda$. In particular, there exists a unique measure of maximal entropy for the flow $X|_\Lambda$.