  Elementary sub-Riemannian geometry on the Heisenberg group H(n) provides a
compact picture of symplectic geometry. Any Hamiltonian diffeomorphism on
$R^{2n}$ lifts to a volume preserving bi-Lipschitz homeomorphisms of H(n), with
the use of its generating function. Any curve of a flow of such homeomorphisms
deviates from horizontality by the Hamiltonian of the flow. From the metric
point of view this means that any such curve has Hausdorff dimension 2 and the
$\mathcal{H}^{2}$ (area) density equal to the Hamiltonian. The non-degeneracy
of the Hofer distance is a direct consequence of this fact.
