Anomalous Anosov flows revisited

TL;DR

This paper explores higher-dimensional Anosov flows, establishing a dichotomy for fiberwise flows over 3D bases and introducing a new surgery method to construct non-transitive flows in odd dimensions.

Contribution

It provides a dichotomy classification for fiberwise Anosov flows and introduces a novel surgery technique for creating non-transitive flows in all odd dimensions.

Findings

Fiberwise Anosov flows are either suspensions or have equal stable and unstable dimensions.

A new surgery method constructs non-transitive Anosov flows in all odd dimensions.

The results deepen understanding of the structure and diversity of higher-dimensional Anosov flows.

Abstract

This paper is devoted to higher dimensional Anosov flows and consists of two parts. In the first part, we investigate fiberwise Anosov flows on affine torus bundles which fiber over 3-dimensional Anosov flows. We provide a dichotomy result for such flows --- they are either suspensions of Anosov diffeomorphisms or the stable and unstable distributions have equal dimensions. In the second part, we give a new surgery type construction of Anosov flows, which yields non-transitive Anosov flows in all odd dimensions.