Invariant submanifolds of conformal symplectic dynamics

Arnaud, Marie-Claude; Fejoz, Jacques

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

We study invariant manifolds of conformal symplectic dynamical systems on a symplectic manifold $(ℳ,\omega )$ of dimension $\ge 4$. We first prove the $\omega$-isotropy of an invariant manifold $𝒩$, assuming the entropy of $𝒩$ is small with respect to the conformality rate. Next, when $(ℳ,\omega )$ is exact and $𝒩$ is isotropic, we show that $𝒩$ must be exact for some choice of the primitive of $\omega$, 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 $(ℳ,\omega )$ de dimension $\ge 4$. Nous montrons d’abord qu’une variété invariante $𝒩$ est $\omega$-isotrope, à supposer que l’entropie de la dynamique restreinte soit petite par rapport au taux de conformalité. Ensuite, quand $(ℳ,\omega )$ est exacte et $𝒩$ isotrope, nous montrons que $𝒩$ est exacte pour un certain choix de primitive de $\omega$, 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 $𝒩$.

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

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