2021/03/21 by Pietro Corvaja, Umberto Zannier, Corvaja, Pietro +1
Computer Science · Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Chaos-based Image/Signal Encryption #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2103.11347
openalex publication_date 2021/03/21 · openalex created_date 2023/10/05 · openalex updated_date 2026/07/28
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,π), 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.