Lorenz-like chaotic attractors revised

TL;DR

This paper reviews recent findings on Lorenz-like attractors, highlighting their expansive dynamics, unique physical measures, and statistical properties such as hitting times and large deviations.

Contribution

It provides a comprehensive update on the dynamical and statistical properties of singular-hyperbolic Lorenz-like attractors, emphasizing new results on their hyperbolicity and measure-theoretic behavior.

Findings

Attractors are expansive and sensitive to initial conditions.

They admit a unique physical measure supported on the entire attractor.

Hitting times follow a logarithm law and large deviations decay exponentially.

Abstract

We describe some recent results on the dynamics of singular-hyperbolic (or Lorenz-like) attractors: attractors in this class are expansive and so sensitive with respect to initial data; they admit a unique physical measure whose support is the whole attractor, which is hyperbolic and the equilibrium state with respect to the center-unstable Jacobian; the hitting time associated to a geometric Lorenz attractor satisfies a logarithm law; the rate of large deviations for the physical measure on the ergodic basin of a geometric Lorenz attractor is exponential.