Existence and smoothness of the stable foliation for sectional hyperbolic attractors

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 $C^{1.278}$.

Corrected estimate for smoothness of stable foliation. Clarification of which results hold for general partially hyperbolic attractors. Some minor typos fixed. Accepted for publication in Bull. London Math. Soc

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