Guaranteed recurrence in any finite system?

Sorry if this is in the wrong forum.

This video puts forward the popular belief that given enough time, a finite system can only exist in finitely many states, and so it must return to its starting position. They extend this to the popular belief that there will be a time in the future where "they make this exact video again", i.e. that we're somehow stuck in an infinite loop, due to there only being a finite number of states in a finite system.

Isn't this just not true? Why couldn't the system evolve so that it reached a cycle? (I.e. like a function that goes 1-2-3-4-5-4-5-4-5), so that states must be repeated eventually, but not necessarily every state. In the example of the balls in the cage (1:55), suppose I have a point in the middle of the cage so that when a ball hits it, it gets stuck. In that case, we still have a finite system that would clearly never return to its original state.

This is probably obvious, I just have always heard the theory of "infinite recurrence" and its mathematical justification, but I guess it doesn't really hold, right?

You're right. Iterating a map from a finite set to itself must reach some point infinitely often but need not reach every point infinitely often (or even at all).

Any other claim without stronger hypotheses is wrong.

That said, there is a sense in which some physical systems are "mixing": trajectories reach every region of states infinitely often. Read about ergodicity.