Given an open book decomposition of a closed contact three manifold with pseudo-Anosov monodromy, connected binding, and fractional Dehn twist coefficient , we construct a Legendrian knot close to the stable foliation of a page, together with a small Legendrian pushoff . When , we apply the techniques of [CH13] to show that the strip Legendrian contact homology of is well-defined and has an exponential growth property. The work [Alv19] then implies that all Reeb vector fields for have positive topological entropy.
On associe à toute décomposition en livre ouvert d’une variété de contact close de dimension , de monodromie pseudo-Anosov, de reliure connexe et de coefficient de Dehn fractionnaire , un nœud legendrien proche du feuilletage stable d’une page accompagné d’un petit translaté legendrien . Lorsque , on applique les techniques de [CH13] pour montrer que l’homologie de contact legendrienne cylindrique de est bien définie et a une propriété de croissance exponentielle. Le travail [Alv19] implique alors que tout champ de Reeb pour a une entropie topologique non nulle.
Accepted:
Published online:
DOI: 10.5802/jep.89
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
@article{JEP_2019__6__119_0, author = {Alves, Marcelo R.R. and Colin, Vincent and Honda, Ko}, 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}, publisher = {Ecole polytechnique}, volume = {6}, year = {2019}, doi = {10.5802/jep.89}, zbl = {1415.57011}, mrnumber = {3915194}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jep.89/} }
TY - JOUR AU - Alves, Marcelo R.R. AU - Colin, Vincent AU - Honda, Ko TI - Topological entropy for Reeb vector fields in dimension three via open book decompositions JO - Journal de l’École polytechnique - Mathématiques PY - 2019 SP - 119 EP - 148 VL - 6 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.89/ DO - 10.5802/jep.89 LA - en ID - JEP_2019__6__119_0 ER -
%0 Journal Article %A Alves, Marcelo R.R. %A Colin, Vincent %A Honda, Ko %T Topological entropy for Reeb vector fields in dimension three via open book decompositions %J Journal de l’École polytechnique - Mathématiques %D 2019 %P 119-148 %V 6 %I Ecole polytechnique %U http://www.numdam.org/articles/10.5802/jep.89/ %R 10.5802/jep.89 %G en %F JEP_2019__6__119_0
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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