Conformally symplectic dynamics, beyond symplectic dynamics – CoSyDy

Our project aims to study conformally symplectic dynamics (CSD) which preserve a symplectic structure up to a constant scaling factor, and its generalizations to conformally symplectic manifolds in all its aspects. In particular, • Which conditions on the dynamics and on the manifold imply the existence of an attractor ? • We will investigate various features of these attractors : their invariant measures, their entropy and the distribution of functions that are defined at almost every point (rotation vectors, asymptotic Maslov indices, Lyapunov exponents etc.) • We will use spectral invariants from symplectic geometry to prove existence and uniqueness results for Lagrangian submanifolds in the attractor. • We will explore the dependence of large families of CSD systems on various parameters, e.g. the scaling factor k. It is believed that the dynamics simplifies when k is small, and one may expect that some information on conservative systems can be obtained by letting k tend to 1.
We will prove rigidity results by using variational symplectic techniques.