TITLE: Surfaces that are 'everywhere accessible' to a randomly positioned Newtonian particle with an arbitrary velocity vector
QUESTION [9 upvotes]: Consider an idealized classical particle confined to a two-dimensional surface that is frictionless.  The particle's initial position on the surface is randomly selected, a nonzero velocity vector is randomly assigned to it, and the direction of the particle's movement changes only at the surface boundaries where perfectly elastic collisions occur (i.e. there is no information loss over time).
My question is - Does there exist such a bounded surface where the probability of the particle visiting any given position at some time 't', P(x,y,t), becomes equal to unity at infinite time?  In other words, no matter where we initialize the particle, and no matter the velocity vector assigned to it, are there surfaces that will always be 'everywhere accessible'?
(Once again, I welcome any help asking this question in a more appropriate manner...)

REPLY [2 votes]: I think the answers are not to the question asked (at least as it is asked). The ergodicity of the geodesic flow (which, by the way holds for all negatively curved surfaces -- a fact surprising not mentioned in any of the above answers) does not mean that a fixed geodesic will hit every point on the surface eventually, but merely that it will become dense (well, more than that, but less than hitting every point). The OP asks for every point to be hit.