Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part I: The dynamically coherent case

2019/08/17 by Thomas Barthelmé, Barthelmé, Thomas, Sérgio R. Fenley +5
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Geometric and Algebraic Topology #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1908.06227

Abstract

We study 3-dimensional dynamically coherent partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the transverse geometry and topology of the center stable and center unstable foliations, and the dynamics within their leaves. We find a structural dichotomy for these foliations, which we use to show that every such diffeomorphism on a hyperbolic or Seifert fibered 3-manifold is leaf conjugate to the time one map of a (topological) Anosov flow. This proves a classification conjecture of Hertz-Hertz-Ures in hyperbolic 3-manifolds and in the homotopy class of the identity of Seifert manifolds.

Related

• Partially hyperbolic diffeormorphisms, ergodicity, and transverse foliations in dimension 3
• Partial hyperbolicity and pseudo-Anosov dynamics
• Partially hyperbolic dynamics and 3-manifold topology
• Partially hyperbolic diffeomorphisms homotopic to the identity on\n 3-manifolds
• Dynamical incoherence for a large class of partially hyperbolic\n diffeomorphisms