  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.
