Simultaneous Laminations

Calegari introduced the laminations $\Lambda^\pm_u$ associated to a universal circle.

We study the laminations $\Lambda^\pm_u$ for pseudo-Anosov orbit space universal circles of taut foliations, on atoroidal three-manifolds.

We prove that $\Lambda^+_u$ and $\Lambda^-_u$ are completely determined by the stable and unstable prelaminations on the boundary of the orbit space.

Then, using a result of Barthelmé, Bonatti, and Mann, we prove that for any action $\rho:\pi_1(M)\to \mathrm{Homeo}^+(S^1)$ coming from an orbit space of a pseudo-Anosov flow, there is a finite collection of lamination pairs such that $(\Lambda^+_u,\Lambda^-_u)$ lies in this collection for any minimal orbit space universal circle whose action is conjugate to $\rho$.