Let’s do an experiment, in which particles move in some space thru time, let’s say in a cube of \(1 m^3\), and you observe the changes of their positions every minute. Observe now a smaller cube inside the first one. Particles may get out of that smaller cube, travel around and may or may not come back. The question is: how do I know if they will come back?
Henri Poncaré, French mathematician born in April, 1854, proved that if the particles are moving in a nice way, most of them will come back. Actually, they return infinitely many times!
Poincaré’s Recurrence Theorem must be one of my favorites theorems, not only for the beauty of the result and how useful it is, but also because the proof is literally three lines that are easy to read.