{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/flows-growth-rates-and-the-veering-polynomial","title":"Flows, growth rates, and the veering polynomial","arxiv_id":"2107.04066","date":"2021-07-08","proceeding":null,"authors":["Michael P. Landry","Yair N. Minsky","Samuel J. Taylor"],"abstract":"For certain pseudo-Anosov flows $\\phi$ on closed $3$-manifolds, unpublished work of Agol--Gu\\'eritaud produces a veering triangulation $\\tau$ on the manifold $M$ obtained by deleting $\\phi$'s singular orbits. We show that $\\tau$ can be realized in $M$ so that its 2-skeleton is positively transverse to $\\phi$, and that the combinatorially defined flow graph $\\Phi$ embedded in $M$ uniformly codes $\\phi$'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 $\\phi$'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 $\\phi$ 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.","url_abs":"https://arxiv.org/abs/2107.04066v3","url_pdf":"https://arxiv.org/pdf/2107.04066v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"flows-growth-rates-and-the-veering-polynomial","repo_url":"https://github.com/henryseg/veering","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}