  We discuss the dynamics of smooth diffeomorphisms of the disc with vanishing
topological entropy which satisfy the mild dissipation property introduced in
[CP]. In particular it contains the H\'enon maps with Jacobian up to 1/4. We
prove that these systems are either (generalized) Morse Smale or infinitely
renormalizable. In particular we prove for this class of diffeomorphisms a
conjecture of Tresser: any diffeomorphism in the interface between the sets of
systems with zero and positive entropy admits doubling cascades. This
generalizes for these surface dynamics a well known consequence of
Sharkovskii's theorem for interval maps.
