Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 8A / c / Solution

The energy sublevel $H\le 1/2$ 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 Poincare recurrence theorem.

Solved by gpt-5.6-sol high.