Homotopical Complexity of a Billiard Flow on the 3D Flat Torus with Two Cylindrical Obstacles

TL;DR

This paper investigates the homotopical behavior of billiard flows on a 3D flat torus with cylindrical obstacles, revealing bounds on escape speeds, convexity of rotation sets, and entropy estimates.

Contribution

It introduces the concept of homotopical rotation vectors for this billiard system and establishes bounds, convexity, and density properties of the rotation sets, advancing understanding of the system's complexity.

Findings

Orbits escape at speeds up to √3.

Any speed up to 1/(√6+2√3) is achievable in any direction.

The rotation set is convex and dense with periodic orbit vectors.

Abstract

We study the homotopical rotation vectors and the homotopical rotation sets for the billiard flow on the unit flat torus with two, disjoint and orthogonal cylindrical scatterers removed from it. The natural habitat for these objects is the infinite cone erected upon the Cantor set $\text{Ends}(G)$ of all "ends" of the hyperbolic group $G=\pi_1(\mathbf{Q})$. An element of $\text{Ends}(G)$ describes the direction in (the Cayley graph of) the group $G$ 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 $e\in\text{Ends}(\pi_1(\mathcal{Q}))$ the escape is feasible with any prescribed speed $s$, $0\leq s\leq\dfrac{1}{\sqrt{6}+2\sqrt{3}}$. 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.