[Paper Review] Partial hyperbolicity and pseudo-Anosov dynamics

Sérgio R. Fenley, Rafaël Potrie|arXiv (Cornell University)|Feb 3, 2021

Mathematical Dynamics and Fractals4 citations

TL;DR

This paper establishes that every closed hyperbolic 3-manifold admitting a partially hyperbolic diffeomorphism also supports an Anosov flow, providing a complete classification of such diffeomorphisms in hyperbolic and Seifert 3-manifolds via their center (branching) foliations and the concept of collapsed Anosov flows. The key contribution is linking dynamical coherence and pseudo-Anosov dynamics to the existence of Anosov flows, resolving a long-standing classification problem in dimension three dynamics.

ABSTRACT

We show that if a hyperbolic 3-manifold admits a partially hyperbolic diffeomorphism then it also admits an Anosov flow. Moreover, we give a complete classification of partially hyperbolic diffeomorphism in hyperbolic 3-manifolds as well as partially hyperbolic diffeomorphisms in Seifert manifolds inducing pseudo-Anosov dynamics in the base. This classification is given in terms of the structure of their center (branching) foliations and the notion of collapsed Anosov flows.

Motivation & Objective

• To resolve the classification problem of partially hyperbolic diffeomorphisms in closed hyperbolic 3-manifolds and Seifert manifolds.
• To establish a topological obstruction for the existence of partially hyperbolic diffeomorphisms by linking them to Anosov flows.
• To introduce and apply the concept of collapsed Anosov flows to unify and generalize existing models, especially in dynamically incoherent settings.
• To characterize the structure of center (branching) foliations and their visual measures in relation to pseudo-Anosov dynamics.

Proposed method

• Use of the theory of partially hyperbolic diffeomorphisms with invariant splitting $TM = E^s \oplus E^c \oplus E^u$ and the $Df^\ell$-contraction/expansion conditions.
• Application of hyperbolic geometry to analyze the action of deck transformations on the ideal boundary of leaves in the universal cover.
• Leveraging results from [BFFP] and [Fen4] on the existence of axes and fixed points at infinity for non-peripheral curves.
• Use of quasi-isometry arguments to show that certain geodesic rays and their images under deck transformations make uniformly bounded angles.
• Employment of the visual measure and coarse geometry to analyze the dynamics of the action on the circle at infinity.
• Reduction to the case of $\mathbb{R}$-covered, uniform foliations via lift and finite cover techniques to apply Theorem 5.6.

Experimental results

Research questions

• RQ1 Under what topological conditions does a closed hyperbolic 3-manifold admit a partially hyperbolic diffeomorphism?
• RQ2 Can the existence of a partially hyperbolic diffeomorphism in a hyperbolic 3-manifold be used to construct an Anosov flow on the same manifold?
• RQ3 How do the structure and dynamics of center (branching) foliations relate to the existence of Anosov flows in Seifert and hyperbolic 3-manifolds?
• RQ4 In what sense do collapsed Anosov flows generalize the notion of leaf conjugacy and time-one maps of Anosov flows?
• RQ5 What is the role of dynamical coherence (or lack thereof) in the classification of partially hyperbolic diffeomorphisms in 3-manifolds?

Key findings

• Every closed hyperbolic 3-manifold admitting a partially hyperbolic diffeomorphism also admits an Anosov flow, establishing a strong topological obstruction.
• Partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds are completely classified via their center foliations and the notion of collapsed Anosov flows.
• In Seifert manifolds with pseudo-Anosov dynamics on the base, the center and strong stable foliations have small visual measure inside the leaves of the center-stable foliation.
• The action of deck transformations on the ideal boundary of leaves has fixed points that are either super-attracting or super-repelling, a key geometric constraint.
• Dynamical incoherence is unavoidable in many cases, as shown by the existence of $\mathbb{R}$-covered, uniform foliations with translation actions on the leaf space.
• The results generalize previous work on time-one maps of Anosov flows and extend the framework to include non-dynamically coherent systems.