Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation

TL;DR

This paper demonstrates that key statistical properties like decay of correlations, SRB measures, and limit theorems hold for singular hyperbolic flows without requiring smooth stable foliations, extending previous results to broader classes.

Contribution

It proves that many statistical properties of singular hyperbolic flows are valid even without the smoothness assumption on the stable foliation, including in higher dimensions.

Findings

Existence of SRB measures for non-smooth flows

Central limit theorems and invariance principles hold

Mixing and rates of mixing are established

Abstract

Over the last 10 years or so, advanced statistical properties, including exponential decay of correlations, have been established for certain classes of singular hyperbolic flows in three dimensions. The results apply in particular to the classical Lorenz attractor. However, many of the proofs rely heavily on the smoothness of the stable foliation for the flow. In this paper, we show that many statistical properties hold for singular hyperbolic flows with no smoothness assumption on the stable foliation. These properties include existence of SRB measures, central limit theorems and associated invariance principles, as well as results on mixing and rates of mixing. The properties hold equally for singular hyperbolic flows in higher dimensions provided the center-unstable subspaces are two-dimensional.