Nonexpanding attractors: Conjugacy to algebraic models and classification in 3-manifolds

Aaron W. Brown

doi:10.3934/jmd.2010.4.517

Article Contents

2010, Volume 4, Issue 3: 517-548. Doi: 10.3934/jmd.2010.4.517

This issue Previous Article Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups Next Article Zygmund strong foliations in higher dimension

Nonexpanding attractors: Conjugacy to algebraic models and classification in 3-manifolds

Aaron W. Brown^1,

1.
Department of Mathematics, Tufts University, Medford, MA 02155

Received: January 2010

Revised: July 2010

Published: September 2010

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Abstract

We prove a result motivated by Williams's classification of expanding attractors and the Franks--Newhouse Theorem on codimension-$1$ Anosov diffeomorphisms: If $\Lambda$ is a topologically mixing hyperbolic attractor such that $\dim\E^u$|_$\Lambda$ = 1, then either $\Lambda$ is expanding or is homeomorphic to a compact abelian group (a toral solenoid). In the latter case, $f$|_$\Lambda$ is conjugate to a group automorphism. As a corollary, we obtain a classification of all $2$-dimensional basic sets in $3$-manifolds. Furthermore, we classify all topologically mixing hyperbolic attractors in $3$-manifolds in terms of the classically studied examples, answering a question of Bonatti in [1].

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Mathematics Subject Classification: Primary: 37C70, 37C15; Secondary: 37D20.

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