Billiard complexity in the hypercube

TL;DR

This paper studies the complexity of billiard trajectories in a hypercube by coding them with face sequences, establishing that the complexity grows polynomially with order approximately n^{3d-3}.

Contribution

It introduces a face-coding method for billiard trajectories in hypercubes and determines the polynomial growth rate of the associated complexity function.

Findings

Complexity grows as n^{3d-3} in hypercube billiards.

Provides a new coding scheme for billiard trajectories.

Establishes the order of magnitude for the complexity function.

Abstract

We consider the billiard map in the hypercube of $\mathbb{R}^d$. We obtain a language by coding the billiard map by the faces of the hypercube. We investigate the complexity function of this language. We prove that $n^{3d-3}$ is the order of magnitude of the complexity.