Volume Preserving Bi-lipschitz Homeomorphisms on the Heisenberg Group
نویسنده
چکیده
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
منابع مشابه
1 The Heisenberg group does not admit a bi - Lipschitz embedding into L 1 after
We show that the Heisenberg group, with its Carnot-Caratheodory metric, does not admit a bi-Lipschitz embedding into L1.
متن کاملCompression bounds for Lipschitz maps from the Heisenberg group to L1
We prove a quantitative bi-Lipschitz nonembedding theorem for the Heisenberg group with its Carnot-Carathéodory metric and apply it to give a lower bound on the integrality gap of the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem.
متن کامل2 00 9 Invariance of Regularity Conditions under Definable , Locally Lipschitz , Weakly Bi - Lipschitz Mappings
In this paper we describe the notion of a weak lipschitzianity of a mapping on a C stratification. We also distinguish a class of regularity conditions that are in some sense invariant under definable, locally Lipschitz and weakly bi-Lipschitz homeomorphisms. This class includes the Whitney (B) condition and the Verdier condition.
متن کاملA Schoenflies Extension Theorem for a Class of Locally Bi-lipschitz Homeomorphisms
In this paper we prove a new version of the Schoenflies extension theorem for collared domains Ω and Ω in Rn: for p ∈ [1, n), locally bi-Lipschitz homeomorphisms from Ω to Ω with locally p-integrable, second-order weak derivatives admit homeomorphic extensions of the same regularity. Moreover, the theorem is essentially sharp. The existence of exotic 7-spheres shows that such extension theorems...
متن کاملMetric Differentiation, Monotonicity and Maps to L
We present a new approach to the infinitesimal structure of Lipschitz maps into L, and as an application, we give an alternative proof of the main theorem of [CK06], that the Heisenberg group does not admit a bi-Lipschitz embedding in L. The proof uses the metric differentiation theorem of [Pau01] and the cut metric description in [CK06] to reduce the nonembedding argument to a classification o...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2002