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

TL;DR

This paper classifies hyperbolic attractors in 3-manifolds, showing they are either expanding or conjugate to algebraic models, and provides a comprehensive classification of 2-dimensional basic sets in 3-manifolds.

Contribution

It establishes a dichotomy for mixing hyperbolic attractors with 1D unstable manifolds and classifies all hyperbolic attractors in 3-manifolds.

Findings

Hyperbolic attractors are either expanding or conjugate to algebraic models.

Classification of all 2-dimensional basic sets in 3-manifolds.

Complete classification of hyperbolic attractors in 3-manifolds.

Abstract

We prove a result motivated by Williams's classification of expanding attractors and the Franks-Newhouse Theorem on codimension-1 Anosov diffeomorphisms: If a mixing hyperbolic attractor has 1-dimensional unstable manifolds then it is either is expanding or is homeomorphic to a compact abelian group (a toral solenoid); in the latter case the dynamics 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 hyperbolic attractors in 3-manifolds in terms of the classically studied examples, answering a question of Bonatti.