Homotopical Complexity of 2D Billiard Orbits

Traditionally, rotation numbers for toroidal billiard flows are defined as the limiting vectors of average displacements per time on trajectory segments. 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 classically 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. The main results of this paper claim that the orbits can only escape to infinity at a speed not exceeding $\sqrt{2}$, and any direction $e\in\text{Ends}(F_2)$ for the escape is feasible with any prescribed speed $s$, $0\leq s\leq \sqrt{2}/2$. This means that the radial upper and lower bounds for the rotation set $R$ are actually pretty close to each other.