Ergodic theory is a theory that comes from Statistical Mechanics. Imagine you have a handful of atoms and a box, now let's say that you line the atoms up along one side of the box and give them all a push perfectly perpendicular to the side (oh, and the sides are perfectly flat). What happens? The atoms keep just bouncing back and forth between those two sides, and large regions of the box never have any atoms in them. If this situation could happen in real life, then Statistical Mechanics would have serious problems! So Ergodic Theory is basically the study of making sure that the atoms in a box bounce all over equally...