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.
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
- Inverses are quasisymmetric
- If f : X → Y is an invertible η-quasisymmetric map as above, then its inverse map is -quasisymmetric, where (t) = 1/η(1/t).
- Quasisymmetric maps preserve relative sizes of sets
- If A and B are subsets of X and A is a subset of B, then
Weakly quasisymmetric maps
A map f:X→Y is said to be H-weakly-quasisymmetric for some H > 0 if for all triples of distinct points x,y,z in X, we have
Not all weakly quasisymmetric maps are quasisymmetric. However, if X is connected and X and Y 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.
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.
Quasisymmetric maps and doubling measures
The real line
Quasisymmetric homeomorphisms of the real line to itself can be characterized in terms of their derivatives. 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
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).
Quasisymmetry and quasiconformality in Euclidean space
Let Ω and Ω´ be open subsets of ℝn. If f : Ω → Ω´ is η-quasisymmetric, then it is also K-quasiconformal, where K > 0 is a constant depending on η.
Conversely, if f : Ω → Ω´ is K-quasiconformal and B(x, 2r) is contained in Ω, then f is η-quasisymmetric on B(x, r), where η depends only on K.
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