Nonexpanding Attractors: Conjugacy to Algebraic Models and\n Classification in 3-Manifolds

2010/05/06 by Aaron Brown, Brown, Aaron W.
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1005.1130

openalex publication_date 2010/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a result motivated by Williams's classification of expanding\nattractors and the Franks-Newhouse Theorem on codimension-1 Anosov\ndiffeomorphisms: If a mixing hyperbolic attractor has 1-dimensional unstable\nmanifolds then it is either is expanding or is homeomorphic to a compact\nabelian group (a toral solenoid); in the latter case the dynamics is conjugate\nto a group automorphism. As a corollary we obtain a classification of all\n2-dimensional basic sets in 3-manifolds. Furthermore we classify all hyperbolic\nattractors in 3-manifolds in terms of the classically studied examples,\nanswering a question of Bonatti.\n

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