Journal of Modern Dynamics (JMD)

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

Pages: 517 - 548, Issue 3, July 2010      doi:10.3934/jmd.2010.4.517

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Aaron W. Brown - Department of Mathematics, Tufts University, Medford, MA 02155, United States (email)

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].

Keywords:  Hyperbolic attractors, solenoids, conjugacy, classification.
Mathematics Subject Classification:  Primary: 37C70, 37C15; Secondary: 37D20.

Received: January 2010;      Revised: July 2010;      Available Online: October 2010.