Orbit growth of contact structures after surgery

Foulon, Patrick; Hasselblatt, Boris; Vaugon, Anne

Annales Henri Lebesgue, Tome 4 (2021), pp. 1103-1141.

Notre étude, à l’intersection des systèmes dynamiques et la géométrie de contact, porte sur les effets de la construction d’une chirurgie de contact adaptée à l’étude des champs de Reeb et sur les effets de l’invariance de l’homologie de contact.

Nous montrons que cette chirurgie de contact produit une complexité dynamique accrue pour tous les flots de Reeb compatibles avec la nouvelle structure de contact. Nous étudions des champs de Reeb Anosov sur des 3-variétés fermées qui ne sont topologiquement orbite-équivalents à aucun flot algébrique, ce qui inclut de nombreux exemples sur des 3-variétés hyperboliques. Notre étude comprend également des résultats de croissance exponentielle dans des cas où, ni le flot obtenu par chirurgie, ni la variété construite ne sont hyperboliques ainsi que des résultats quand le flot d’origine est périodique. Ce travail démontre pleinement, dans ce cadre, la pertinence de l’homologie contact pour analyser la dynamique des champs de Reeb.

This work is at the intersection of dynamical systems and contact geometry, and it focuses on the effects of a contact surgery adapted to the study of Reeb fields and on the effects of invariance of contact homology.

We show that this contact surgery produces an increased dynamical complexity for all Reeb flows compatible with the new contact structure. We study Reeb Anosov fields on closed 3-manifolds that are not topologically orbit-equivalent to any algebraic flow; this includes many examples on hyperbolic 3-manifolds. Our study also includes results of exponential growth in cases where neither the flow nor the manifold obtained by surgery is hyperbolic, as well as results when the original flow is periodic. This work fully demonstrates, in this context, the relevance of contact homology to the analysis of the dynamics of Reeb fields.

Classification : 37D20, 37D40, 53E50, 57K33
Mots-clés : Anosov flow, 3-manifold, contact structure, contact flow, Reeb flow, surgery, contact homology

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