Consider a square pool table $1\times1$. We hit a ball, and observe what happens. There is no friction and this system obeys law of reflections. I am intrested where the ball is going to hit.
Now let's take a point $x_0$, where the ball hit the wall for the first time. Consider a segment of wall $I$. $|I| = \epsilon$ and it has $x_0$ as the middle point.
Is this true, that the ball will be hitting the interval $I$ infinitely often? How to (dis)prove this?
I have an intuition, that i should use Poincaré recurrence theorem.
The function here is $T(p_1,\alpha_1) = (p_2,\alpha_2)$.
$p_1$-is the previous point where the ball hit, $p_2$ is the next point.
$\alpha_1$ - is an angle between the wall where the previous point lies and the dircetion to the next point, we define $\alpha_2$ similarly
But the question is how to prove that this function preserves Lebesgue measure.
Regards
This is true, but you don't need Poincare recurrence theorem.
As a hint, try to prove that either the trajectory of the ball is periodic (in this case, it will hit exactly $x_0$ an infinity of times), either it is dense in the square (in this case, it will not hit $x_0$ again, but it will hit $I$ an infinity of times).