Topological entropy for Reeb vector fields in dimension three via open book decompositions

Marcelo R.R. Alves; Vincent Colin; Ko Honda

doi:10.5802/jep.89

Topological entropy for Reeb vector fields in dimension three via open book decompositions
[Entropie topologique des champs de Reeb en dimension

3

via les livres ouverts]

Marcelo R.R. Alves ¹ ; Vincent Colin ² ; Ko Honda ³

¹ Département de Mathématique, Université Libre de Bruxelles, CP 218, Boulevard du Triomphe, B-1050 Bruxelles, Belgique
² Université de Nantes, UMR 6629 du CNRS 44322 Nantes, France
³ University of California, Los Angeles Los Angeles, CA 90095, USA

Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 119-148.

Résumé
Abstract

On associe à toute décomposition en livre ouvert d’une variété de contact close $(M, ξ)$ de dimension $3$ , de monodromie pseudo-Anosov, de reliure connexe et de coefficient de Dehn fractionnaire $c = k / n$ , un nœud legendrien $Λ$ proche du feuilletage stable d’une page accompagné d’un petit translaté legendrien $\hat{Λ}$ . Lorsque $k \geq 5$ , on applique les techniques de [CH13] pour montrer que l’homologie de contact legendrienne cylindrique de $Λ \to \hat{Λ}$ est bien définie et a une propriété de croissance exponentielle. Le travail [Alv19] implique alors que tout champ de Reeb pour $ξ$ a une entropie topologique non nulle.

Given an open book decomposition of a closed contact three manifold $(M, ξ)$ with pseudo-Anosov monodromy, connected binding, and fractional Dehn twist coefficient $c = k / n$ , we construct a Legendrian knot $Λ$ close to the stable foliation of a page, together with a small Legendrian pushoff $\hat{Λ}$ . When $k \geq 5$ , we apply the techniques of [CH13] to show that the strip Legendrian contact homology of $Λ \to \hat{Λ}$ is well-defined and has an exponential growth property. The work [Alv19] then implies that all Reeb vector fields for $ξ$ have positive topological entropy.

Reçu le : 2017-05-21
Accepté le : 2019-01-15
Publié le : 2019-02-10

MR Zbl

DOI : 10.5802/jep.89

Classification : 57M50, 37B40, 53C15
Keywords: Topological entropy, contact structure, open book decomposition, mapping class group, Reeb dynamics, pseudo-Anosov, contact homology
Mot clés : Entropie topologique, structure de contact, livre ouvert, groupe de difféotopie, dynamique de Reeb, pseudo-Anosov, homologie de contact

Affiliations des auteurs :

Marcelo R.R. Alves ¹ ; Vincent Colin ² ; Ko Honda ³

¹ Département de Mathématique, Université Libre de Bruxelles, CP 218, Boulevard du Triomphe, B-1050 Bruxelles, Belgique
² Université de Nantes, UMR 6629 du CNRS 44322 Nantes, France
³ University of California, Los Angeles Los Angeles, CA 90095, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Marcelo R.R. Alves and Vincent Colin and Ko Honda},
     title = {Topological entropy for {Reeb} vector fields in dimension three via open book decompositions},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {119--148},
     publisher = {\'Ecole polytechnique},
     volume = {6},
     year = {2019},
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     zbl = {1415.57011},
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}

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Marcelo R.R. Alves; Vincent Colin; Ko Honda. Topological entropy for Reeb vector fields in dimension three via open book decompositions. Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 119-148. doi : 10.5802/jep.89. https://jep.centre-mersenne.org/articles/10.5802/jep.89/

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