The space of (contact) Anosov flows on 3-manifolds

TL;DR

This paper explores the topology of the space of Anosov flows on 3-manifolds, revealing infinite connected components and properties of contact Anosov flows, with implications for their classification and stability.

Contribution

It characterizes the topology of the space of Anosov vector fields on hyperbolic surface bundles and analyzes the stability of contact Anosov flows under time changes.

Findings

Countably infinite connected components of Anosov vector fields.

Each component is not simply connected.

Time changes of contact Anosov flows form a $C^1$-open subset.

Abstract

The first half of this paper is concerned with the topology of the space $\AAA(M)$ of (not necessarily contact) Anosov vector fields on the unit tangent bundle $M$ of closed oriented hyperbolic surfaces $\Sigma$. We show that there are countably infinite connected components of $\AAA(M)$, each of which is not simply connected. In the second part, we study contact Anosov flows. We show in particular that the time changes of contact Anosov flows form a $C^1$-open subset of the space of the Anosov flows which leave a particular $C^\infty$ volume form invariant, if the ambiant manifold is a rational homology sphere.