Rigidity of pseudo-Anosov flows transverse to R-covered foliations

TL;DR

This paper proves that for R-covered foliations, there are at most two pseudo-Anosov flows transverse to them, and characterizes the case when two such flows exist using the universal circle.

Contribution

It establishes a classification of pseudo-Anosov flows transverse to R-covered foliations, linking their existence to the foliation's conjugacy to stable foliations of R-covered Anosov flows.

Findings

At most two pseudo-Anosov flows transverse to R-covered foliations.

If two such flows exist, the foliation is weakly conjugate to a stable foliation of an R-covered Anosov flow.

Uses the universal circle to R-covered foliations in the proof.

Abstract

A foliation is R-covered if the leaf space in the universal cover is homeomorphic to the real numbers. We show that, up to topological conjugacy, there are at most two pseudo-Anosov flows transverse to such a foliation. If there are two, then the foliation is weakly conjugate to the the stable foliation of an R-covered Anosov flow. The proof uses the universal circle to R-covered foliations.