Invariant submanifolds of conformal symplectic dynamics

Marie-Claude Arnaud; Jacques Fejoz

doi:10.5802/jep.252

¹Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, F-75006 Paris, France & Institut universitaire de France
²Université Paris Dauphine, CEREMADE, Place du Maréchal de Lattre de Tassigny, 75016 Paris, France & Observatoire de Paris, IMCCE, 77, avenue Denfert Rochereau, 75014 Paris, France

Journal de l’École polytechnique — Mathématiques, Volume 11 (2024), pp. 159-185

Abstract (Original language)
Résumé

We study invariant manifolds of conformal symplectic dynamical systems on a symplectic manifold $(ℳ, ω)$ of dimension $\geq 4$ . We first prove the $ω$ -isotropy of an invariant manifold $𝒩$ , assuming the entropy of $𝒩$ is small with respect to the conformality rate. Next, when $(ℳ, ω)$ is exact and $𝒩$ is isotropic, we show that $𝒩$ must be exact for some choice of the primitive of $ω$ , under the condition that the dynamics acts trivially on the cohomology of degree $1$ of $𝒩$ . The conclusion partially extends if a one-sided orbit of $𝒩$ has compact closure. We eventually describe some conditions showing the uniqueness of $𝒩$ .

Nous étudions les variétés invariantes des systèmes dynamiques conformes symplectiques sur une variété symplectique $(ℳ, ω)$ de dimension $\geq 4$ . Nous montrons d’abord qu’une variété invariante $𝒩$ est $ω$ -isotrope, à supposer que l’entropie de la dynamique restreinte soit petite par rapport au taux de conformalité. Ensuite, quand $(ℳ, ω)$ est exacte et $𝒩$ isotrope, nous montrons que $𝒩$ est exacte pour un certain choix de primitive de $ω$ , sous la condition que la dynamique agit trivialement sur la cohomologie de degré $1$ de $𝒩$ . La conclusion se généralise partiellement si une demi-orbite de $𝒩$ est d’adhérence compacte. Enfin, nous décrivons des conditions montrant l’unicité de $𝒩$ .

Article information
Cite this article

Received: 2022-03-19
Accepted: 2023-12-06
Published online: 2024-01-09

MR Zbl

DOI: 10.5802/jep.252

Classification: 37C05, 37J39, 38A35
Keywords: Conformal symplectic dynamics, isotropy, entropy, exactness, Lagrangian submanifold, invariant manifold
Mots-clés : Dynamique conformément symplectique, isotropie, exactitude, variétés lagrangiennes, variétés invariantes

Author's affiliations:

Marie-Claude Arnaud ¹ ; Jacques Fejoz ²

¹ Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, F-75006 Paris, France & Institut universitaire de France
² Université Paris Dauphine, CEREMADE, Place du Maréchal de Lattre de Tassigny, 75016 Paris, France & Observatoire de Paris, IMCCE, 77, avenue Denfert Rochereau, 75014 Paris, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

Marie-Claude Arnaud; Jacques Fejoz. Invariant submanifolds of conformal symplectic dynamics. Journal de l’École polytechnique — Mathématiques, Volume 11 (2024), pp. 159-185. doi: 10.5802/jep.252

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