Smooth conjugacy classes of 3d Axiom A flows

Anna Florio (1, 2) , Martin Leguil (3, 4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Université Paris Dauphine-PSL
3 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352
4 UPJV - Université de Picardie Jules Verne

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.

Domaines: Systèmes dynamiques [math.DS]

Anna Florio, Martin Leguil. Smooth conjugacy classes of 3d Axiom A flows. 2022. ⟨hal-03852329⟩