2021/10/11 by Arnaud, Marie-Claude, Jacques Féjoz, Fejoz, Jacques · 2 citations
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Quantum chaos and dynamical systems
paper · doi:10.48550/arxiv.2110.05062
openalex publication_date 2021/10/11 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We study invariant manifolds of conformal symplectic dynamical systems on a symplectic manifold (M, ω) of dimension ≥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 ω-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, ω) is exact and N is isotropic, we also 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 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.