*Speaker. *Jacques Fejoz *Date.* 7.05.24 at 15:30

*Abstract.* We study invariant submanifolds of conformal symplectic dynamical systems on a symplectic manifold. This class of systems is the 1-dimensional extension of symplectic dynamical systems for which the symplectic form is transformed colinearly to itself, and contains mechanical systems whose friction is proportional to the velocity. It is well know that ergodic quasiperiodic invariant tori must be isotropic. Yet, there exist examples of compact invariant submanifolds under a conform symplectic dynamics which are not isotropic. We determine how the isotropy of an invariant manifold 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. This is a joint work with Marie-Claude Arnaud.