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Partially hyperbolic diffeomorphisms homotopic to the identity in dimension $3$, II: Branching foliations

Thomas Barthelmé, Sérgio R Fenley, Steven Frankel and Rafael Potrie

Geometry & Topology 27 (2023) 3095–3181

We study 3–dimensional partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the geometry and dynamics of Burago and Ivanov’s center stable and center unstable branching foliations. This extends our previous study of the true foliations that appear in the dynamically coherent case. We complete the classification of such diffeomorphisms in Seifert fibered manifolds. In hyperbolic manifolds, we show that any such diffeomorphism is either dynamically coherent and has a power that is a discretized Anosov flow, or is of a new potential class called a double translation.

partial hyperbolicity, 3–manifolds, foliations
Mathematical Subject Classification
Primary: 37D30, 57R30, 37C15, 57M50, 37D20
Received: 7 August 2020
Revised: 4 February 2022
Accepted: 13 March 2022
Published: 9 November 2023
Proposed: David Gabai
Seconded: Leonid Polterovich, David Fisher
Thomas Barthelmé
Department of Mathematics and Statistics
Queen’s University
Kingston, ON
Sérgio R Fenley
Department of Mathematics
Florida State University
Tallahassee, FL
United States
Steven Frankel
Department of Mathematics
Washington University in St. Louis
St Louis, MO
United States
Rafael Potrie
Facultad de Ciencias – Centro de Matemática
Universidad de la República

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