  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 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.
