Existence and smoothness of the stable foliation for sectional hyperbolic attractors

2016/04/30 by V. Araújo, I. Melbourne
Mathematics · #math.DS

paper · pdf · doi:10.1112/blms.12037

published as Bull. London Math. Soc. 49 (2017) 351-367 · 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

arxiv created 2017/01/26 · arxiv updated 2017/12/06

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 C¹ which is crucial for recent results on exponential decay of correlations. In fact the foliation is at least C^1.278.

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• Multidimensional non-uniform hyperbolicity, robust exponential mixing and the basin problem