This should be true in a more general setting, but for simplicity consider billiards that are connected, compact subsets of the plane with boundary $C^2$ except at finitely many points. A ball (or a ray of light) rolls inside, going in straight lines, and upon collision with the wall, the orbit is reflected.
It is intuitive that a statement like the following is true:
For almost every billiard, there exists an orbit that is dense everywhere inside it.
However, as far as I know this is still open. In fact, the last thing I heard was that it had just been proven for the case in which the billiard was an obtuse triangle with certain restrictions (but I have since forgotten the source, unfortunately).
Question: What is the current status of the problem?
Thank you!
Clarification The question is not about rectangular billiard tables, but in general about the balls rolling in more general shapes. 'Almost always' would then have to be given a meaning within the space of curves. (In fact, the problem is trivial in rectangles because an orbit with irrational slope will do.) Also, this is not about having dense families of orbits, but a single orbit that is dense in the billiard.
I think the way 'almost always' should be defined is by requiring some generic property to hold. Think, for example, of the definition R. Abraham give of bumpy metrics.
Best Answer
Do you mean to ask whether the trajectories in almost all cases (in {shapes X trajectories} are dense in the set of {positions, directions} on the table, or just in the set positions? The first question seems more natural to me; the answer is no: If there are two convex portions of the boundary curve pointing toward each other, they're like convex mirrors, they tend to focus. For open sets of shapes and distances, the return map from the tangent line bundle along a mirror back to itself has eigenvalues of the first derivative a pair of complex conjugate points on the unit circle. Because of the KAM theory, there are typically rings of positive measure consisting of invariant circles for the return map. The orbits of these rings under the billiard flow enclose an open set in phase space.
Another physical example of this effect what happens when you wind something like kite string around a flat object, perhaps a piece of cardboard or a board. The string tends to build up in the middle, and once a bulge gets started, the configuration is stable---the string prefers to wind back and forth across the bulge. (Note that the shortest paths of winding string follow geodesics, which are the same as trajectories as billiards on a table of the same shape).
Even when the KAM situation isn't obvious from the geometry of the table, I think experimental evidence shows that it's commonplace. There are known constructions of Riemannian metrics on $S^2$ with ergodic geodesic flow, but they took a long time before someone found them (sorry, I don't remember the reference). Similarly, I think it's tricky to find examples of simply-connected billiard tables with smooth boundary that are ergodic: you somehow have to systematically eliminate the KAM phenomenon. It's much easier if the table either has angles, or is multiply-connected with two or more obstacles in the middle (so that doubling it produces a surface of negative Euler characteristic).
It's not obvious to me how to use this phenomenon to capture all trajectories that pass through a particular point, but maybe that's not really the most natural question: after all, in a game of billiards, direction and position both matter.