Measure of recurring trajectories on a billiard

Question

My question follows from the solutions of a simple problem: on a pool table with no friction, no holes, using point-like billiard balls, and following the laws of reflection, can you hit a ball positioned in the point $P$ so that its trajectory will never pass at $P$ again?

I resolved that in 2 ways:

1. you unfold the pool table and draw another one on the edge where the balls hit first and so on, so the ball's trajectory is a straight line: we know that if the slope of this line is an irrational number, than the balls will never pass on the point $P$ again, because if so, an irrational number could be expressed as a fraction of 2 integers (the number of the unfolding for each edge), so the subset of the irrational numbers is the solution.

2. It's possible to demonstrate that the pool table satisfies the hypotheses of Poincarè recurrence theorem: "If $S$ is a space with measure $\mu$ and $T:S \to S$ is a measure-preserving transformation, then for any set with positive measure $B⊆S$, the subset $A⊆B$ of points that never recur to $B$ has measure zero."(https://digitalcommons.coastal.edu/honors-theses/23/), and so there exist a subset of points that will never recur, but this subset's measure is $0$. So we have solved the problem this way too.

But we know that the measure of the subset of the irrational numbers is not $0$. So how is that possible?

Answer 1

First of all you have to develop more your second affirmation. Indeed what is $S$ in your example? It should be the space $S := \{(x,r)\}$ where $x$ is a point on the boundary of your billiard and $r$ is a slope. Then your application $T$ give the point and the slope just after the next rebound. Then you can show that there is a invariant measure $\mu$ on it which is continuous with Lesbegue.

Finally your set $B$ you are considering is $B := \{ (x,r), x=P \}$ which have measure $0$. And there is no contradiction with the fact that the subset $A \subset B$ with no recur is also a $0$ set.

If you would have take $B := \{ (x,r), x \in I \}$ with $I$ an non empty interval of the boundary of the billard, then you would have $\mu(B) \ne 0$, and the set of non-recur would be different (certainly not the irrational slope) but sill with measure $0$.