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

TL;DR

This paper proves finiteness theorems for certain periodic trajectories in elliptical billiards, linking geometric billiard patterns with arithmetic and dynamical conjectures, and explores their implications through algebraic and Diophantine methods.

Contribution

It establishes new finiteness results for elliptical billiard trajectories related to arithmetic conjectures and connects these to a variant of the dynamical Mordell-Lang conjecture.

Findings

Finiteness of periodic trajectories with fixed initial conditions.

Finiteness of billiard shots leading to a target ball in a hole.

Explicit example involving Diophantine equations in algebraic tori.

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.