Polygonal billiards and uniform distribution According to this article in Wikipedia: A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed. Billiard dynamical systems are Hamiltonian idealizations of the game of billiards, but where the region contained by the boundary can have shapes other than rectangular and even be multidimensional. 
My question is motivated by a videogame I've been playing lately, which can be seen at http://www.youtube.com/watch?v=LLLmfwxNJYU.
Essentially, the "physics" of the game involves several billiards in a polygon, and the player has to slash off pieces of the polygon while avoiding the billiards.  Also, the removed piece has to be void of billiards.
I've been assuming that the distribution of billiards is in the long run uniform, in some hand waving sense.  That's assuming that initial distributions and velocities are random.  Is that true, or can the polygon be shaped in various ways to make that distribution non-uniform?  In other words, are certain regions of a polygon are more likely to be void than other regions?
(I believe that a term like "ergodic" applies to this, but I'm not confident using it). 

The very difficult questions that has emerged over the past 20 years in billiard theory is the following one :
is there a non-periodic orbit that is not everywhere dense in the billiard. 
a few years ago, Galperin thought that he had found one but an other mathematician, Tokarsky, proved that Galperin's orbit was actually a periodic one. So this is still an open question for specialists but it seems from computer simulations and theoretical results (see Masur, Boshernitzan, Tabachnikov...) that a non periodic orbit will be everywhere dense in the polygon. 
