Advances in symplectic billiards

In this PhD thesis, we present recent results on symplectic billiards. This class of mathematical billiards was introduced by P. Albers and S. Tabachnikov as a billiard dynamics having an area, rather than a length, as generating function. The thesis addresses three different problems related to this dynamical system. The first concerns integrability: we establish a Bialy–Mironov type result for symplectic billiards. The second question is related to the inverse problem, and asks whether the spectral properties of symplectic billiard dynamics allow us to reconstruct the shape of the domain. In this direction, we provide two results: one regarding the expansion of Mather’s β-function, and the other concerning the rigidity property of the so-called area spectrum. Finally, in the last chapter, we turn to the setting of conformal symplectic dynamics and propose a model of dissipative symplectic billiards. This system has a global attractor whose topological and dynamical complexity varies both in terms 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.