Balancedness Constants of Words Generated By Billiards in The Hypercube.

RESUMEN     In this talk, I will consider words generated by coding the trajectory of a ball inside a hypercubic billiard table. Balancedness is a combinatorial notion related to the dynamical concept of discrepancy, which characterizes square billiard words. It is then natural to ask: what is the imbalance (i.e., the optimal balancedness constant) of hypercubic billiard words in higher dimensions?

After reviewing the partial results obtained by Vuillon in 2003, I will present a complete characterization of the imbalances of hypercubic billiard words generated by the trajectory of a ball launched with an initial momentum whose coordinates are rationally independent. In a second part, I will also examine the more intricate case, where the coordinates of the momentum are rationally dependent.

Posted on Apr 24, 2025 in Dynamical Systems, Seminars