Papers › Flows, growth rates, and the veering polynomial
Flows, growth rates, and the veering polynomial
Michael P. Landry, Yair N. Minsky, Samuel J. Taylor
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For certain pseudo-Anosov flows ϕ on closed $3$-manifolds, unpublished work of Agol--Gu\'eritaud produces a veering triangulation τ on the manifold M obtained by deleting ϕ's singular orbits. We show that τ can be realized in M so that its 2-skeleton is positively transverse to ϕ, and that the combinatorially defined flow graph Φ embedded in M uniformly codes ϕ's orbits in a precise sense. Together with these facts we use a modified version of the veering polynomial, previously introduced by the authors, to compute the growth rates of ϕ's closed orbits after cutting M along certain transverse surfaces, thereby generalizing work of McMullen in the fibered setting. These results are new even in the case where the transverse surface represents a class in the boundary of a fibered cone of M. Our work can be used to study the flow ϕ on the original closed manifold. Applications include counting growth rates of closed orbits after cutting along closed transverse surfaces, defining a continuous, convex entropy function on the `positive' cone in H¹ of the cut-open manifold, and answering a question of Leininger about the closure of the set of all stretch factors arising as monodromies within a single fibered cone of a $3$-manifold. This last application connects to the study of endperiodic automorphisms of infinite-type surfaces and the growth rates of their periodic points.
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