Homotopical Rotation Numbers of 2D Billiards

Abstract

Traditionally, rotation numbers for toroidal billiard flows are defined as the limiting vectors of average displacements per time on trajectory segments. Naturally, these creatures are living in the (commutative) vector space $\real^n$, if the toroidal billiard is given on the flat $n$-torus. The billard trajectories, being curves, oftentimes getting very close to closed loops, quite naturally define elements of the fundamental group of the billiard table. The simplest non-trivial fundamental group obtained this way belongs to the classical Sinai billiard, i.e., the billiard flow on the 2-torus with a single, convex obstacle removed. This fundamental group is known to be the group $\textbf{F}_2$ freely generated by two elements, which is a heavily noncommutative, hyperbolic group in Gromov's sense. We define the homotopical rotation number and the homotopical rotation set for this model, and provide lower and upper estimates for the latter one, along with checking the validity of classicaly expected properties, like the density (in the homotopical rotation set) of the homotopical rotation numbers of periodic orbits. The natural habitat for these objects is the infinite cone erected upon the Cantor set $\text{Ends}(\textbf{F}_2)$ of all ``ends'' of the hyperbolic group $\textbf{F}_2$. An element of $\text{Ends}(\textbf{F}_2)$ describes the direction in (the Cayley graph of) the group $\textbf{F}_2$ in which the considered trajectory escapes to infinity, whereas the height function $t$ ($t \ge 0$) of the cone gives us the average speed at which this escape takes place.