Uniqueness of equilibrium states for sectional-hyperbolic flows, including the classical Lorenz attractor

Abstract

It has long been conjectured that the classical Lorenz attractor supports a unique measure of maximal entropy. In this talk, we will 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. we will prove that on every compact manifold with dimension at least three, there exists a $C^1$-open and dense family of vector fields that includes the classical Lorenz attractor (when $\dim M=3$), such that if the point masses at singularities are not equilibrium states, then there exists a unique equilibrium state. In particular, there exists a unique measure of maximal entropy. This is joint work with Maria Jose Pacifico and Jiagang Yang.

Speaker

I am currently a Visiting Assistant Professor at Michigan State University. I obtained my Ph.D. from the University of Southern California in 2015. I had a postdoc position at Instituto de Matematica Pura e Aplicada (IMPA) in Brazil, and the University of Oklahoma, USA.

Starting in July 2023, I will be an assistant professor at Wake Forest University.

Research Interests:

Smooth ergodic theory for flows with singularities.

Statistical properties of partially hyperbolic systems: existence and uniqueness of equilibrium measures, large deviation properties, decay of correlations.

Limit theorems in deterministic and random dynamical systems: escape rates, hitting times statistics, extreme value distributions, central limit theorem, and other limit theorems.

Entropy theory and its relations with topological structures.