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

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Let (XdX) and (YdY) 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 xyz of distinct points in X, we have

 

Basic properties

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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

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Weakly quasisymmetric maps

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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

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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

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The real line

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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

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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

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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

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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

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Let (XdX) and (YdY) 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 xyzt of distinct points in X, we have

 

See also

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References

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  1. ^ Heinonen, Juha (2001). Lectures on Analysis on Metric Spaces. Universitext. New York: Springer-Verlag. pp. x+140. ISBN 978-0-387-95104-1.
  2. ^ 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.
  3. ^ Beurling, A.; Ahlfors, L. (1956). "The boundary correspondence under quasiconformal mappings". Acta Math. 96: 125–142. doi:10.1007/bf02392360.
  4. ^ 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.
  5. ^ Buyalo, Sergei; Schroeder, Viktor (2007). Elements of Asymptotic Geometry. EMS Monographs in Mathematics. American Mathematical Society. p. 209. ISBN 978-3-03719-036-4.