[Paper Review] Partially hyperbolic diffeomorphisms homotopic to the identity on 3-manifolds

Thomas Barthelmé, Sérgio R. Fenley+2 more|arXiv (Cornell University)|Dec 31, 2017

Mathematical Dynamics and Fractals13 references6 citations

TL;DR

This research announcement classifies partially hyperbolic diffeomorphisms on 3-manifolds homotopic to the identity by showing they are dynamically coherent and conjugate to discretized Anosov flows when the manifold is Seifert fibered or hyperbolic, using foliation theory and rigidity arguments. The key result establishes that such diffeomorphisms are either time-1 maps of Anosov flows or exhibit double translation behavior, with the latter ruled out in hyperbolic manifolds under certain conditions.

ABSTRACT

We announce some results towards the classification of partially hyperbolic diffeomorphisms on 3-manifolds, and outline the proofs in the case when the diffeomorphism is dynamically coherent. Detailed proofs are long and technical and will appear later.

Motivation & Objective

• To classify partially hyperbolic diffeomorphisms on 3-manifolds homotopic to the identity using an inside-out approach based on foliation structure.
• To resolve the obstruction of non-integrable central stable/unstable bundles by proving dynamical coherence under geometric constraints.
• To show that such diffeomorphisms are conjugate to discretized Anosov flows in Seifert and hyperbolic 3-manifolds.
• To rule out the existence of non-dynamically coherent examples with double translation behavior in hyperbolic manifolds.
• To extend classification results to general irreducible 3-manifolds, particularly graph manifolds, using JSJ decomposition and pseudo-Anosov-like flows.

Proposed method

• Use of foliation theory to analyze the integrability of central stable and unstable bundles, particularly through weak foliations.
• Application of the theory of R-covered Anosov flows and their reversing maps to construct candidate double translation examples.
• Employment of leaf conjugacy and minimality of branching foliations to deduce dynamical coherence.
• Use of the Seifert fibration structure to construct a 'Seifert trick' that forces dynamical coherence in Seifert manifolds.
• Rigidity arguments based on the non-existence of C¹ reversing maps (e.g., η) for non-suspension Anosov flows to rule out double translation in hyperbolic manifolds.
• Extension of results to general irreducible 3-manifolds via JSJ decomposition and conjectured existence of regulating pseudo-Anosov-like flows in atoroidal pieces.

Experimental results

Research questions

• RQ1 Can every partially hyperbolic diffeomorphism on a 3-manifold homotopic to the identity be conjugate to a discretized Anosov flow?
• RQ2 Under what conditions does dynamical coherence follow from geometric and topological constraints in 3-manifolds?
• RQ3 Is the existence of a double translation behavior—where both weak foliations are preserved but not uniquely integrable—possible in hyperbolic 3-manifolds?
• RQ4 Can the Seifert fibration structure be used to force dynamical coherence in Seifert manifolds?
• RQ5 What role does the JSJ decomposition play in extending classification results to general irreducible 3-manifolds, especially graph manifolds?

Key findings

• In Seifert fibered 3-manifolds, every partially hyperbolic diffeomorphism homotopic to the identity is dynamically coherent and conjugate to a discretized Anosov flow.
• In hyperbolic 3-manifolds, the existence of a double translation example (non-dynamically coherent) is ruled out under the assumption that the reversing map η is not C¹, which holds for non-suspension Anosov flows.
• The failure of the reversing map η to be C¹ in non-suspension R-covered Anosov flows prevents the construction of a C¹ double translation diffeomorphism.
• The theory of weak foliations and their minimality leads to dynamical coherence after proving that branching leaves and their intersections are fixed.
• In general irreducible 3-manifolds, the classification remains open, but an analogue of the main theorem is expected in atoroidal pieces with pseudo-Anosov-like flows.
• The case of graph manifolds remains unresolved due to the absence of atoroidal pieces and a center in the fundamental group, requiring new techniques.