Simultaneous universal circles and continuous extension

Abstract

Fenley proved that any foliation almost transverse to a quasigeodesic pseudo-Anosov flow in a closed atoroidal 3-manifold has the continuous extension property, meaning the inclusions of leaves into the universal cover continuously extend to their ideal boundaries. This article gives an alternate proof of an upgraded version of this: the associated Cannon-Thurston map for the flow, constructed by Frankel and Fenley, organizes all of the leafwise continuous extensions. The proof uses the fact that the boundary of the flowspace is naturally a universal circle for the foliation.