Robust Exponential Mixing and Convergence to Equilibrium for Singular Hyperbolic Attracting Sets

TL;DR

This paper proves that singular hyperbolic attracting sets exhibit robust exponential mixing and convergence to equilibrium, extending previous results to more complex systems with multiple singularities and physical measures.

Contribution

It generalizes exponential mixing and convergence results to singular-hyperbolic sets with multiple singularities and physical measures in higher-dimensional manifolds.

Findings

Exponential mixing holds for physical measures in these systems.

Convergence to equilibrium occurs exponentially for a broad class of vector fields.

Results apply to systems with multiple Lorenz- or non-Lorenz-like singularities.

Abstract

We extend results on robust exponential mixing for geometric Lorenz attractors, with a dense orbit and a unique singularity, to singular-hyperbolic attracting sets with any number of (either Lorenz- or non-Lorenz-like) singularities and finitely many ergodic physical/SRB invariant probability measures, whose basins cover a full Lebesgue measure subset of the trapping region of the attracting set. We obtain exponential mixing for any physical probability measure supported in the trapping region and also exponential convergence to equilibrium, for a $C^2$ open subset of vector fields in any $d$-dimensional compact manifold ($d\ge3$).