Rainbow meanders and Cartesian billiards

In this paper we relate several objects from quite diverse areas of mathematics. Closed meanders are the configurations which arise when one or several disjoint closed Jordan curves in the plane intersect the horizontal axis transversely. The question of their connectivity also arises when evaluating traces in Temperley-Lieb algebras. The variant of open meanders is closely related to the detailed dynamics of Sturm global attractors, i.e. the global attractors of parabolic PDEs in one space dimension; see the groundbreaking work of Fusco and Rocha [FuRo91]. Cartesian billiards have their corners located on the integer Cartesian grid with corner angles of ±90 degrees. Billiard paths are at angles of ±45 degrees with the boundaries and reflect at half-integer coordinates. We indicate and explore some close connections between these seemingly quite different objects.