Solution
= Solution
The energy sublevel $H\le1/2$ is compact and has finite phase volume because both $p$ and $q$ are bounded. If all $\Phi_{nT}(U)$ were disjoint, volume preservation would put infinitely many equal positive volumes inside it, impossible. Thus two iterates intersect. Pulling the intersection back shows a point of $U$ returns to $U$; applying the argument to arbitrarily small neighbourhoods gives <Poincaré recurrence theorem>.