  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.
