Topological entropy for Reeb vector fields in dimension three via open book decompositions
Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 119-148.

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 Λ ^. When k5, we apply the techniques of [CH13] to show that the strip Legendrian contact homology of ΛΛ ^ 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.

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 Λ ^. Lorsque k5, on applique les techniques de [CH13] pour montrer que l’homologie de contact legendrienne cylindrique de ΛΛ ^ 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.

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Accepted:
Published online:
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
Alves, Marcelo R.R. 1; Colin, Vincent 2; Honda, Ko 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
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     title = {Topological entropy for {Reeb} vector fields in dimension three via open book decompositions},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique - Math\'ematiques},
     pages = {119--148},
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Alves, Marcelo R.R.; Colin, Vincent; Honda, Ko. Topological entropy for Reeb vector fields in dimension three via open book decompositions. Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 119-148. doi : 10.5802/jep.89. http://www.numdam.org/articles/10.5802/jep.89/

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