Baker, baker give me recurrence as fast as you can

The Baker's map is a simple dynamical system that has deterministic chaos. The map is defined on a square. Cut the dough in half vertically then stack them on top of each other after kneading each piece. It is illustrated below.

The exponential growth between two different points is especially easy to see for two points that are just separated in the horizontal direction and not in the vertical direction at all. The distance between them doubles every time, at least until they end up on different sides of the vertical cut, then its not so easy to say what happens.

It doubles the horizontal direction and halves the vertical direction, so it will preserve area for every rectangle, which is a basis. So extending this preservation to the sigma algebra they generate, you see that this map preserves Lebesgue measure.

A picture of the chaotic nature.

By composition, we get an action of this map on functions on the square. It is unitary on the Hilbert space of square integrable ones. Once we figure out this operator's spectrum, we will be able to exactly say what this map does to arbitrary square integrable functions. Express the function in the diagonalized basis then use the eigenvalues to determine the action on each summand.

Each eigenvalue will have some power N that takes it as close to 1 as you want. If you want an entire square integrable function to come back to look like itself, take the least common multiple of each N of the summands that contribute a significant amount. Some analysis will give you exactly which of the components you'll need to consider. Let's call that applying the baker map M times.

The punchline to the above is you can get a function that is arbitrary close after M iterations. This demonstrates Poincare recurrence for this dynamical system. So it starts out looking more and more disordered, but then it comes back to the original state or at least close enough.

If you deleted a bit when it was in it's chaotic state, then you would be screwed. Think like if you forgot the states in the Black Hole and didn't remember to keep track of all the Hawking radiation.