Quasi-Anosov diffeomorphisms of 3-manifolds

TL;DR

This paper classifies quasi-Anosov diffeomorphisms on 3-manifolds, describing their topology and dynamics, and provides a comprehensive structure theorem for these systems.

Contribution

It solves Hirsch's problem for hyperbolic invariant sets that are 3-manifolds, detailing their topological types and dynamical classifications.

Findings

Orientable case: manifolds are connected sums of tori and handles.

Non-orientable case: manifolds are connected sums quotiented by involutions.

Dynamics are classified as connected sums of DA-diffeomorphisms with involutions.

Abstract

In 1969, Hirsch posed the following problem: given a diffeomorphism, and a compact invariant hyperbolic set, describe its topology and restricted dynamics. We solve the problem where the hyperbolic invariant set is a closed 3-manifold: if the manifold is orientable, then it is a connected sum of tori and handles; otherwise it is a connected sum of tori and handles quotiented by involutions. The dynamics of the diffeomorphisms restricted to these manifolds, called quasi-Anosov diffeomorphisms, is also classified: it is the connected sum of DA-diffeomorphisms, quotiented by commuting involutions.