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Invariant Submanifolds of conformal Symplectic Dynamics


We study invariant manifolds of conformal symplectic dynamical systems on a symplectic manifold $(\cM,\omega)$ of dimension $\geq 4$. This class of systems is the $1$-dimensional extension of symplectic dynamical systems for which the symplectic form is transformed colinearly to itself. In this context, we first examine how the $\omega$-isotropy of an invariant manifold $\cN$ relates to the entropy of the dynamics it carries. Central to our study is Yomdin's inequality, and a refinement obtained using { that the local entropies have no effect transversally to the characteristic foliation of $\cN$.} When $(\cM,\omega)$ is exact and $\cN$ is isotropic, %and isotopic to a graph, we also show that $\cN$ 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 $\cN$. The conclusion partially extends to the case when $\cN$ has a compact one-sided orbit. We eventually prove the uniqueness of invariant submanifolds $\cN$ when $\cM$ is a cotangent bundle, provided that the dynamics is isotopic to the identity among Hamiltonian diffeomorphisms. In the case of the cotangent bundle of the torus, a theorem of Shelukhin allows us to conclude that $\cN$ is unique even among submanifolds with compact orbits.
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hal-03372201 , version 1 (09-10-2021)



Marie-Claude Arnaud, Jacques Fejoz. Invariant Submanifolds of conformal Symplectic Dynamics. 2021. ⟨hal-03372201⟩
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