Abstract: 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 ϕ 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 ϕ. Our primary tool is a way to use the flow space of ϕ to parameterize the circle bundle at infinity of ϕ’s invariant foliations.

Abstract: Let g : L → L be an atoroidal, endperiodic map on an infinite type surface L with no boundary and finitely many ends, each of which is accumulated by genus. By work of Landry, Minsky, and Taylor, g is isotopic to a spun pseudo-Anosov map f. We show that spun pseudo-Anosov maps minimize the number of periodic points of period n for sufficiently high n over all maps in their homotopy class, strengthening a theorem of Landry, Minsky, and Taylor. We also show that the same theorem holds for atoroidal Handel--Miller maps when one only considers periodic points that lie in the intersection of the stable and unstable laminations. Furthermore, we show via example that spun-pseudo Anosov and Handel--Miller maps do not always minimize the number of periodic points of low period.