Denseness of robust exponential mixing for singular-hyperbolic attracting sets

TL;DR

This paper proves that in high-dimensional manifolds, a generic set of vector fields with singular-hyperbolic attracting sets exhibit robust exponential mixing for physical measures, even under small perturbations.

Contribution

It establishes the denseness and robustness of exponential mixing in singular-hyperbolic attracting sets for a broad class of vector fields.

Findings

Existence of a dense subset of vector fields with exponential mixing.

Robustness of exponential mixing under small perturbations.

Applicability to high-dimensional manifolds with singular-hyperbolic sets.

Abstract

There exists a $C^2$-open and $C^1$-dense subset of vector fields exhibiting singular-hyperbolic attracting sets (with codimension-two stable bundle), in any $d$-dimensional compact manifold ($d\ge3$), which mix exponentiallu with respect to any physical/SRB invariant probability measure. More precisely, we show that given any connected singular-hyperbolic attracting set for a $C^2$-vector field $X$, there exists a $C^1$-close multiple of $X$ of class $C^2$, generating a topologically equivalent flow, which is robustly exponentially mixing with respect to any physical measure for all vector fields in a $C^2$ neighborhood. That is, every singular-hyperbolic attracting set mixes exponentially with respect to its physical measures modulo an arbitrarily small change in the speed of the flow.