Homotopical Complexity of a 3D Billiard Flow

TL;DR

This paper investigates the homotopical rotation vectors and sets for a 3D billiard flow on a torus with cylindrical scatterers, revealing bounds on escape speeds, convexity of rotation vectors, and entropy estimates.

Contribution

It introduces the concept of homotopical rotation sets for this billiard flow and establishes bounds, convexity, and density results that were not previously known.

Findings

Orbits escape to infinity at speeds up to √3.

Any prescribed speed up to 1/3 is feasible in any direction.

The set of rotation vectors of periodic orbits is dense in the constructible rotation set.

Abstract

In this paper we study the homotopical rotation vectors and the homotopical rotation sets for the billiard flow on the unit flat torus with three, mutually intersecting and mutually orthogonal cylindrical scatterers removed from it. The natural habitat for these objects is the infinite cone erected upon the Cantor set $\text{Ends}(\textbf{F}_3)$ of all "ends" of the hyperbolic group $\textbf{F}_3=\pi_1(\mathbf{Q})$. An element of $\text{Ends}(\textbf{F}_3)$ describes the direction in (the Cayley graph of) the group $\textbf{F}_3$ in which the considered trajectory escapes to infinity, whereas the height function $s$ ($s\ge 0$) of the cone gives us the average speed at which this escape takes place. The main results of this paper claim that the orbits can only escape to infinity at a speed not exceeding $\sqrt{3}$, and in any direction $e\in\text{Ends}(\mathbf{F}_3)$ the escape is feasible with any prescribed speed $s$, $0\leq s\leq 1/3$. This means that the radial upper and lower bounds for the rotation set $R$ are actually pretty close to each other. Furthermore, we prove the convexity of the set $AR$ of constructible rotation vectors, and that the set of rotation vectors of periodic orbits is dense in $AR$. We also provide effective lower and upper bounds for the topological entropy of the studied billiard flow.