Flows, growth rates, and the veering polynomial

• Published in: Ergodic theory and dynamical systems Vol. 43; no. 9; pp. 3026 - 3107
• Main Authors: LANDRY, MICHAEL P., MINSKY, YAIR N., TAYLOR, SAMUEL J.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.09.2023
• Subjects: Automorphisms; Continuity (mathematics); Cutting; Entropy; Manifolds; Orbits; Original Article; Polynomials; Triangulation; 57M60; topological dynamics; 37D20; low-dimensional dynamics; symbolic dynamics; 37C27; 57K32; group actions
• ISSN: 0143-3857; 1469-4417
• DOI: 10.1017/etds.2022.63

For a pseudo-Anosov flow $\varphi $ without perfect fits on a closed $3$ -manifold, Agol–Guéritaud produce a veering triangulation $\tau $ on the manifold M obtained by deleting the singular orbits of $\varphi $ . We show that $\tau $ can be realized in M so that its 2-skeleton is positively transverse to $\varphi $ , and that the combinatorially defined flow graph $\Phi $ embedded in M uniformly codes the orbits of $\varphi $ 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 the closed orbits of $\varphi $ after cutting M along certain transverse surfaces, thereby generalizing the 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 $\varphi $ 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 $H^1$ 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.