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Flows, growth rates, and the veering polynomial

Part of: Smooth dynamical systems: general theory Low-dimensional topology Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  04 October 2022

MICHAEL P. LANDRY Open the ORCID record for MICHAEL P. LANDRY [Opens in a new window] ,
YAIR N. MINSKY Open the ORCID record for YAIR N. MINSKY [Opens in a new window]  and
SAMUEL J. TAYLOR Open the ORCID record for SAMUEL J. TAYLOR [Opens in a new window]
MICHAEL P. LANDRY*
Affiliation:
Department of Mathematics and Statistics, Washington University in Saint Louis, St. Louis, MO 63130, USA
YAIR N. MINSKY
Affiliation:
Department of Mathematics, Yale University, New Haven, CT 06520, USA (e-mail: yair.minsky@yale.edu)
SAMUEL J. TAYLOR
Affiliation:
Department of Mathematics, Temple University, Philadelphia, PA 19122, USA (e-mail: samuel.taylor@temple.edu)
*
e-mail: mlandry@wustl.edu
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Abstract

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.

Keywords

low-dimensional dynamicstopological dynamicssymbolic dynamicsgroup actions

MSC classification

Primary: 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37C27: Periodic orbits of vector fields and flows 57M60: Group actions in low dimensions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 9 , September 2023 , pp. 3026 - 3107
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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