Finiteness theorems on elliptical billiards and a variant of the Dynamical Mordell-Lang Conjecture

Abstract

We offer some theorems, mainly of finiteness, for certain patterns in elliptical billiards, related to periodic trajectories. For instance, if two players hit a ball at a given position and with directions forming a fixed angle in $(0,\pi)$, there are only finitely many cases for both trajectories being periodic. Another instance is the finiteness of the billiard shots which send a given ball into another one so that this falls eventually in a hole. These results have their origin in `relative' cases of the Manin-Mumford conjecture, and constitute instances of how arithmetical content may affect chaotic behaviour (in billiards). We shall also interpret the statements through a variant of the dynamical Mordell-Lang conjecture. In turn, this embraces cases which, somewhat surprisingly, can be treated (only) by completely different methods compared to the former; here we shall offer an explicit example related to diophantine equations in algebraic tori.