This is a question in a rather well investigated subject of which I know very little and I have a hard time "translating" the general results available. Let me also say that I got interested in this question after a conversation with a friend about placements of cell phone towers. (It is too long a story to include here.)


Here's the problem. $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\bsP}{\boldsymbol{P}}$ $\newcommand{\bR}{\mathbb{R}}$ Fix a positive integer $n$ and let $\bsP\_n$ denote the regular $n$-gon centered at the origin and inscribed in the unit circle.


The billiard in the title refers to the motion of a unit speed photon inside $\bsP\_n$ assuming the edges of $\bsP\_n$ are perfect mirrors. For every $x$ in the interior of $\bsP\_n$, any unit vector identified with a $\theta\in S^1$ and any $T>0$ we denote by $N\_T(x,\theta)$ the number of "bounces-off-mirrors" during the time interval $[0,T]$ of a photon that starts its motion at $x$ in the direction $\theta$. The question involves the behavior of $N\_T(x,\theta)$ for large $T$.


I believe that there exists a measurable function $\beta\_n:\bsP\_n\times S^1\to [0,\infty)$ such that for almost all $(x,\theta)\in\bsP\_n\times S^1$ we have


$$\lim\_{T\to\infty}\frac{1}{T}N\_T(x,\theta)=\beta\_n(x,\theta).$$


The questions involve the proportionality constants $\beta\_n(x,\theta)$.


1. Has anyone computed *explicitly* the function $\beta\_n(x,\theta)$?
2. Has anyone computed *explicitly* the average


$$\bar{\beta}\_n(x):=\frac{1}{2\pi} \int\_0^{2\pi}\beta\_n(x,\theta) d\theta ?$$


(I am inclined to believe that $\bar{\beta}\_n(x)$ is constant.)


Above, the cases $n=3,6$ are of particular interest in the original cell phone tower problem.


Finally, has anyone investigated the behavior of $\beta\_n(x,\theta)$ as $n\to\infty$? 


Thanks.



A question about billiards
The answer is given by the general theory of rational billiard flows (i.e., those on polygons whose angles are rational multiples of $\pi$). On such a polygon $Q$ the tangent vectors to any given orbit are parallel to a finite set of unit vectors, so that the orbits with initial direction $\theta$ lie on an invariant surface $M\_\theta$ which consists of a finite number of copies of $Q$, one for each potential direction of an orbit with initial direction $\theta$. According to a theorem of Kerckhoff, Masur and Smillie (<http://www.ams.org/mathscinet-getitem?mr=855297> or the announcement <http://www.ams.org/mathscinet-getitem?mr=799797>), for almost all $\theta$ the flow on $M\_\theta$ is uniquely ergodic. It implies that for a.e. $\theta$ the projection to $Q$ of every orbit with initial direction $\theta$ is uniformly distributed in $Q$ (with respect to the Lebesgue measure).


Therefore, $1/\beta\_n(x,\theta)$ coincides, for a.e. $\theta$ and any $x\in Q$, with the average length of segments in billiard orbits, i.e., by the ergodic theorem, with the average of the function $L(x,\theta)$ (the distance between the points where the line issued from $x$ in the direction $\theta$ intersects the boundary of $Q$) with respect to the Lebesgue measure on $M\_\theta$, which only depends on $\theta$ and should be quite easy to caclulate explicitly in the case of regular polygons.


