  Let I be an open interval, M be a real manifold, T*M its cotangent bundle and
\Phi={\phi_t}, t in I, a homogeneous Hamiltonian isotopy of T*M defined outside
the zero-section. Let \Lambda be the conic Lagrangian submanifold associated
with \Phi (\Lambda is a subset of T*M x T*M x T*I). We prove the existence and
unicity of a sheaf K on MxMxI whose microsupport is contained in the union of
\Lambda and the zero-section and whose restriction to t=0 is the constant sheaf
on the diagonal of MxM. We give applications of this result to problems of non
displaceability in contact and symplectic topology. In particular we prove that
some strong Morse inequalities are stable by Hamiltonian isotopies and we also
give results of non displaceability for positive isotopies in the contact
setting. In this new version we suppress one hypothesis in the main theorem and
we extend the result of non displaceability for positive isotopies.
