Birkhoff attractors of dissipative billiards

We consider a particle moving within some convex planar billiard according to a modified reflection law, where collisions become inelastic; more precisely, at each (non-orthogonal) collision with the boundary, the (unoriented) outgoing angle of reflection is strictly smaller than the incoming angle of incidence, both being measured with respect to the normal. The resulting dissipative billiard map has a global attractor. In a joint work with A. Florio and O. Bernardi, we study the topological and dynamical complexity of an invariant subset of this attractor, the so-called Birkhoff attractor, whose study goes back to Birkhoff, Charpentier, and more recently, the work of Le Calvez. We show that for a convex billiard with pinched curvature, if the dissipation is strong enough, then the Birkhoff attractor is « simple » (a normally contracted manifold) and the dynamics is of Morse-Smale type; on the contrary, we show that if the dissipation is mild, then the Birkhoff attractor is topologically « complicated » (an indecomposable continuum) and has rich dynamics (rotation set with non-empty interior, presence of horseshoes…).