An obstruction to fiberwise Anosov flows over 3-dimensional Anosov flows

TL;DR

This paper investigates conditions under which three-dimensional Anosov flows can serve as bases for fiberwise Anosov flows, identifying obstructions related to periodic orbits and classifying certain flows as suspensions or geodesic flows.

Contribution

It establishes non-existence results for fiberwise Anosov flows over certain base flows and classifies R-covered Anosov flows as suspensions or geodesic flows.

Findings

No fiberwise Anosov flow exists if the base has infinitely many periodic orbits in the same free homotopy class.

R-covered Anosov flows as bases are orbit equivalent to suspensions or geodesic flows.

Abstract

We study obstructions preventing a three-dimensional Anosov flow from serving as the base of a fiberwise Anosov flow. We prove a non-existence result if the base flow admits infinitely many periodic orbits in the same free homotopy class. We get as a corollary that any R-covered Anosov flow serving as the base of a fiberwise Anosov flow is orbit equivalent to a suspension or a geodesic flow.