Thursday, 19th March 2026
12:00 am, Seminari MA, Dept IMAE
Santiago Barbieri (Universitat de Girona): "Existence and nonexistence of invariant curves of coin billiards" (joint work with A. Clarke)
Abstract: In this seminar, I will consider the coin billiard introduced by M. Bialy. It is a modification of the classical billiard, obtained as the return map of a nonsmooth geodesic flow on a cylinder that has homeomorphic copies of a classical billiard on the top and on the bottom (a coin). The return dynamics is described by a map 𝑇 of the annulus 𝔸 =𝕋 ×(0,𝜋). Together with A. Clarke, I proved the following three main theorems: in two different scenarios (when the height of the coin is small, or when the coin is near-circular) there is a family of KAM curves close to, but not accumulating on, the boundary 𝜕𝔸; for any noncircular coin, if the height of the coin is sufficiently large, there is a neighbourhood of 𝜕𝔸 through which there passes no homotopically nontrivial (essential) invariant curve; and the only coin billiard for which the phase space 𝔸 is foliated by essential invariant curves is the circular one. These results provide partial answers to questions of Bialy. Finally, I will describe the results of some numerical experiments on the elliptical coin billiard.