Quasisymmetric map
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In mathematics, a quasisymmetric homeomorphism between metric spaces is a map that generalizes bi-Lipschitz maps. While bi-Lipschitz maps shrink or expand the diameter of a set by no more than a multiplicative factor, quasisymmetric maps satisfy the weaker geometric property that they preserve the relative sizes of sets: if two sets A and B have diameters t and are no more than distance t apart, then the ratio of their sizes changes by no more than a multiplicative constant. These maps are also related to quasiconformal maps, since in many circumstances they are in fact equivalent.[1]
Definition
Let (X, dX) and (Y, dY) be two metric spaces. A homeomorphism f:X → Y is said to be η-quasisymmetric if there is an increasing function η : [0, ∞) → [0, ∞) such that for any triple x, y, z of distinct points in X, we have
Basic properties
- Inverses are quasisymmetric
- If f : X → Y is an invertible η-quasisymmetric map as above, then its inverse map is -quasisymmetric, where
- Quasisymmetric maps preserve relative sizes of sets
- If and are subsets of and is a subset of , then
Examples
Weakly quasisymmetric maps
A map f:X→Y is said to be H-weakly-quasisymmetric for some if for all triples of distinct points in , then
Not all weakly quasisymmetric maps are quasisymmetric. However, if is connected and and are doubling, then all weakly quasisymmetric maps are quasisymmetric. The appeal of this result is that proving weak-quasisymmetry is much easier than proving quasisymmetry directly, and in many natural settings the two notions are equivalent.
δ-monotone maps
A monotone map f:H → H on a Hilbert space H is δ-monotone if for all x and y in H,
To grasp what this condition means geometrically, suppose f(0) = 0 and consider the above estimate when y = 0. Then it implies that the angle between the vector x and its image f(x) stays between 0 and arccos δ < π/2.
These maps are quasisymmetric, although they are a much narrower subclass of quasisymmetric maps. For example, while a general quasisymmetric map in the complex plane could map the real line to a set of Hausdorff dimension strictly greater than one, a δ-monotone will always map the real line to a rotated graph of a Lipschitz function L:ℝ → ℝ.[2]
Doubling measures
The real line
Quasisymmetric homeomorphisms of the real line to itself can be characterized in terms of their derivatives.[3] An increasing homeomorphism f:ℝ → ℝ is quasisymmetric if and only if there is a constant C > 0 and a doubling measure μ on the real line such that
Euclidean space
An analogous result holds in Euclidean space. Suppose C = 0 and we rewrite the above equation for f as
Writing it this way, we can attempt to define a map using this same integral, but instead integrate (what is now a vector valued integrand) over ℝn: if μ is a doubling measure on ℝn and
then the map
is quasisymmetric (in fact, it is δ-monotone for some δ depending on the measure μ).[4]
Quasisymmetry and quasiconformality in Euclidean space
Let and be open subsets of ℝn. If f : Ω → Ω´ is η-quasisymmetric, then it is also K-quasiconformal, where is a constant depending on .
Conversely, if f : Ω → Ω´ is K-quasiconformal and is contained in , then is η-quasisymmetric on , where depends only on .
Quasi-Möbius maps
A related but weaker condition is the notion of quasi-Möbius maps where instead of the ratio only the cross-ratio is considered:[5]
Definition
Let (X, dX) and (Y, dY) be two metric spaces and let η : [0, ∞) → [0, ∞) be an increasing function. An η-quasi-Möbius homeomorphism f:X → Y is a homeomorphism for which for every quadruple x, y, z, t of distinct points in X, we have
See also
References
- ^ Heinonen, Juha (2001). Lectures on Analysis on Metric Spaces. Universitext. New York: Springer-Verlag. pp. x+140. ISBN 978-0-387-95104-1.
- ^ Kovalev, Leonid V. (2007). "Quasiconformal geometry of monotone mappings". Journal of the London Mathematical Society. 75 (2): 391–408. CiteSeerX 10.1.1.194.2458. doi:10.1112/jlms/jdm008.
- ^ Beurling, A.; Ahlfors, L. (1956). "The boundary correspondence under quasiconformal mappings". Acta Math. 96: 125–142. doi:10.1007/bf02392360.
- ^ Kovalev, Leonid; Maldonado, Diego; Wu, Jang-Mei (2007). "Doubling measures, monotonicity, and quasiconformality". Math. Z. 257 (3): 525–545. arXiv:math/0611110. doi:10.1007/s00209-007-0132-5. S2CID 119716883.
- ^ Buyalo, Sergei; Schroeder, Viktor (2007). Elements of Asymptotic Geometry. EMS Monographs in Mathematics. American Mathematical Society. p. 209. ISBN 978-3-03719-036-4.