  We study the dynamics of dissipative billiard maps within planar convex
domains. Such maps have a global attractor. We are interested in the
topological and dynamical complexity of the attractor, in terms both of the
geometry of the billiard table and of the strength of the dissipation. We focus
on the study of an invariant subset of the attractor, the so-called Birkhoff
attractor. On the one hand, we show that for a generic convex table with
"pinched" curvature, the Birkhoff attractor is a normally contracted manifold
when the dissipation is strong. On the other hand, for a mild dissipation, we
prove that generically the Birkhoff attractor is complicated, both from the
topological and the dynamical point of view.
