Universal circles for Anosov foliations

Thurston introduced the notion of a universal circle associated to a taut foliation of a $3$-manifold as a way of organizing the ideal circle boundaries of its leaves into a single circle action. Calegari–Dunfield proved that every taut foliation of an atoroidal $3$-manifold $M$ has a universal circle, but the uniqueness (or lack-thereof) of this structure remains rather mysterious. In this paper, we consider the foliations associated to an Anosov flow $\phi$ on $M$, showing that several constructions of a universal circle in the literature are typically distinct. Moreover, the underlying action of the Calegari–Dunfield leftmost universal circle is generally not even conjugate to the universal circle arising from the boundary of the flow space of $\phi$. Our primary tool is a way to use the flow space of $\phi$ to parameterize the circle bundle at infinity of $\phi$’s invariant foliations.