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

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 Hölder continuous functions on a sectional-hyperbolic attractor $\hat{I}›$. 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 $\hat{I}›$. In particular, there exists a unique measure of maximal entropy for the flow $X{|}_{\hat{I}}›$.

96 pages, 10 figures. This version contains the classical Lorenz attractor as an example. To appear on Duke Math. J