Symplectic billiards for pairs of polygons

We introduce symplectic billiards for pairs of possibly non-convex polygons.

After establishing basic properties, we give several criteria on pairs of polygons for the symplectic billiard map to be fully periodic, i.e. $\textit{every}$ orbit is periodic.

The first fully periodic examples were discovered by Albers-Tabachnikov [AT18] and Albers-Banhatti-Sadlo-Schwartz-Tabachnikov in [ABS+25].

Our criteria allow us to construct a plethora of new examples.

Moreover, we provide an example of a pair of polygons where the symplectic billiard map is fully periodic while having orbits of arbitrarily large period.

After giving a class of examples which provably have isolated periodic orbits (and are thus not fully periodic) we exhibit the first example without any periodic orbits at all.

It is open whether having no periodic orbits at all is possible in the single polygon setting.

Finally, we prove that if one replaces polygons by smooth, strictly convex curves then there are always infinitely many periodic orbits.