Volume preserving bi-Lipschitz homeomorphisms on the Heisenberg group

Details Volume preserving bi-Lipschitz homeomorphisms on the Heisenberg group Volume preserving bi-Lipschitz homeomorphisms on the Heisenberg group

2002-05-04

arxiv.org/abs/math/0205039v2

Mathematics

Symplectic Geometry

Scientific paper

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.

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