Lee M. Goswick, Nándor Simányi
Homotopical Rotation Numbers of 2D Billiards
Preprint series:
ESI preprints
- MSC:
- 11R52 Quaternion and other division algebras: arithmetic, zeta functions
- 52C07 Lattices and convex bodies in $n$ dimensions, See Also {11H06, 11H31, 11P21}
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 $\BR^n$, 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.
Keywords: Rotation number, rotation set, hyperbolic billiards, trajectory, orbit segment, fundamental group, Cayley graph, ideal boundary