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

2022/09/22 by Pacifico, Maria Jose, Yang, Fan, Yang, Jiagang · 2 citations
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2209.10784

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 Λ. We prove that in a C¹-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 Λ. In particular, there exists a unique measure of maximal entropy for the flow X|_Λ.

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• Construction of equilibrium states revisited
• Unique equilibrium states for some partially hyperbolic diffeomorphisms with dominated splittings

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