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We study 3-dimensional dynamically coherent partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the transverse geometry and topology of the center-stable and center-unstable foliations, and the dynamics within their leaves. We find a structural dichotomy for these foliations, which we use to show that every such diffeomorphism on a hyper- bolic or Seifert-fibered 3-manifold is leaf-conjugate to the time-one map of a (topological) Anosov flow. This proves a classification conjecture of Hertz– Hertz–Ures in hyperbolic 3-manifolds and in the homotopy class of the identity of Seifert manifolds.
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Flows, growth rates, and the veering polynomial
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.
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- PAR ID:
- 10417704
- Publisher / Repository:
- EMS Press
- Date Published:
- Journal Name:
- Ergodic Theory and Dynamical Systems
- ISSN:
- 0143-3857
- Page Range / eLocation ID:
- 1 to 82
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/etds.2022.63
