Volume Preserving Bi-lipschitz Homeomorphisms on the Heisenberg Group

The use of sub-Riemannian geometry on the Heisenberg group H(n) provides a compact picture of symplectic geometry. Any Hamiltonian diffeomorphism on R 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 H (area) density equal to the Hamiltonian. The nondegeneracy of the Hofer distance is a direct consequence of this fact. MSC 2000: 53D35, 53C17