A baker’s dozen of problems

Given a convex plane domain, the billiard ball map sends the incoming ray (the trajectory of the billiard ball) to the outgoing one: the law of reflection is “the angle of incidence equals the angle of reflection”. Consider two nested convex domains. Then one has two billiard ball maps, T1 and T2, acting on the oriented lines that intersect both domains. If the domains are bounded by confocal ellipses, then the respective billiard ball maps commute; see, e.g., [32]. Assume that the two maps commute: T1 ◦ T2 = T2 ◦ T1. Conjecture: the two domains are bounded by confocal ellipses. For outer (a.k.a. dual) billiards, an analogous fact is proved in [29]. For piece-wise analytic billiards, this conjecture was proved by A. Glutsyuk [18]. Of course, this problem has a multi-dimensional version, open both for inner and outer billiards.