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

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7 Scopus citations

Abstract

We prove a result motivated by Williams's classification of expanding attractors and the Franks-Newhouse Theorem on codimension-1 Anosov diffeomorphisms: If Λ is a topologically mixing hyperbolic attractor such that dim Eu{up harpoon right}Λ = 1, then either Λ is expanding or is homeomorphic to a compact abelian group (a toral solenoid). In the latter case, f {up harpoon right}Λ 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].

Original languageEnglish (US)
Pages (from-to)517-548
Number of pages32
JournalJournal of Modern Dynamics
Volume4
Issue number3
DOIs
StatePublished - Jul 2010

Keywords

  • Classification
  • Conjugacy
  • Hyperbolic attractors
  • Solenoids

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

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