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Smooth conjugacy classes of 3d Axiom A flows

Florio, Anna; Leguil, Martin (2022), Smooth conjugacy classes of 3d Axiom A flows. https://basepub.dauphine.psl.eu/handle/123456789/23763

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Type
Document de travail / Working paper
Date
2022
Series title
Cahier de recherche CEREMADE, Université Paris Dauphine-PSL
Published in
Paris
Pages
37
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Author(s)
Florio, Anna
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Leguil, Martin
Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 [LAMFA]
Abstract (EN)
We show a rigidity result for 3-dimensional contact Axiom A flows: given two 3D contact Axiom A flows Φ_1, Φ_2 whose restrictions Φ_1|Λ_1 , Φ_2|Λ_2 to basic sets Λ_1, Λ_2 are orbit equivalent, we prove that if periodic orbits in correspondence have the same length, then the conjugacy is as regular as the flows and respects the contact structure, extending a previous result due to Feldman-Ornstein [23]. Some of the ideas are reminiscent of the work of Otal [54]. As an application, we show that the billiard maps of two open dispersing billiards without eclipse and with the same marked length spectrum are smoothly conjugated.