Invariant submanifolds of conformal symplectic dynamics

TL;DR

This paper investigates the properties and uniqueness of invariant submanifolds in conformal symplectic systems, extending classical symplectic results and applying entropy inequalities to understand their structure.

Contribution

It introduces new insights into invariant submanifolds of conformal symplectic systems, including conditions for isotropy, exactness, and uniqueness in cotangent bundles.

Findings

Invariant submanifolds relate to entropy via Yomdin's inequality.

Isotropic invariant manifolds are exact under certain conditions.

Uniqueness of invariant submanifolds in cotangent bundles is established.

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.