Rigidity of pseudo-Anosov flows transverse to $\mathbb{R}$-covered foliations

Abstract

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