Existence and smoothness of the stable foliation for sectional hyperbolic attractors

TL;DR

This paper proves the existence and smoothness of stable foliations for sectional hyperbolic attractors, including the Lorenz system, with implications for decay of correlations.

Contribution

It establishes verifiable bunching conditions ensuring smooth stable foliations for sectional hyperbolic attractors, improving understanding of their geometric structure.

Findings

Stable foliation exists in a neighborhood of sectional hyperbolic attractors.

Under bunching conditions, the foliation is at least $C^{1.278}$.

The stable foliation for the Lorenz system is smoother than $C^1$.

Abstract

We prove the existence of a contracting invariant topological foliation in a full neighborhood for partially hyperbolic attractors. Under certain bunching conditions it can then be shown that this stable foliation is smooth. Specialising to sectional hyperbolic attractors, we give a verifiable condition for bunching. In particular, we show that the stable foliation for the classical Lorenz equation (and nearby vector fields) is better than $C^1$ which is crucial for recent results on exponential decay of correlations. In fact the foliation is at least $C^{1.278}$.