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).