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Invariant submanifolds of conformal symplectic dynamics

Authors :
Arnaud, Marie-Claude
Fejoz, Jacques
Publication Year :
2021

Abstract

We study invariant manifolds of conformal symplectic dynamical systems on a symplectic manifold (M, $\omega$) of dimension $\ge$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 N 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 N. When (M, $\omega$) is exact and N is isotropic, we also 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 to the case when N has a relatively compact one-sided orbit. We eventually prove the uniqueness of invariant submanifolds N when M 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 N is unique even among submanifolds with compact orbits.

Subjects

Subjects :
Mathematics - Dynamical Systems

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2110.05062
Document Type :
Working Paper