Quasisymmetric map explained

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, d_X) and (Y, d_Y) be two metric spaces. A homeomorphism f:X → Y is said to be η-quasisymmetric if there is an increasing function η : [0,&nbsp;∞)&nbsp;→&nbsp;[0,&nbsp;∞) such that for any triple ''x'',&nbsp;''y'',&nbsp;''z'' of distinct points in ''X'', we have :<math> \frac{d_Y(f(x),f(y))}{d_{Y}(f(x),f(z))} \leq \eta\left(\frac{d_X(x,y)}{d_X(x,z)}\right). </math> ==Basic properties== ; Inverses are quasisymmetric : If ''f''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;''Y'' is an invertible ''η''-quasisymmetric map as above, then its inverse map is <math>\eta'</math>-quasisymmetric, where <math display=inline>\eta'(t) = 1/\eta^{-1}(1/t).</math> ; Quasisymmetric maps preserve relative sizes of sets : If <math>A</math> and <math>B</math> are subsets of <math>X</math> and <math>A</math> is a subset of <math>B</math>, then :: <math> \frac{\eta^{-1}(\frac{\operatorname{diam} B}{\operatorname{diam} A})}{2}\leq \frac{\operatorname{diam}f(B)}{\operatorname{diam}f(A)}\leq 2\eta\left(\frac{\operatorname{diam} B}{\operatorname{diam}A}\right).</math> ==Examples== ===Weakly quasisymmetric maps=== A map ''f:X→Y'' is said to be '''H-weakly-quasisymmetric''' for some <math>H>0</math> if for all triples of distinct points <math>x,y,z</math> in <math>X</math>, then :<math> |f(x)-f(y)|\leq H|f(x)-f(z)|\;\;\;\text{ whenever }\;\;\; |x-y|\leq |x-z|</math> Not all weakly quasisymmetric maps are quasisymmetric. However, if <math>X</math> is [[Connected space|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.

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,

\langle f(x) - f(y),x - y\rangle\geq \delta |f(x) - f(y)| ⋅ |x - y|.

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

f(x) = C + \int ^{x} _{0} d\mu(t).

Euclidean space

An analogous result holds in Euclidean space. Suppose C = 0 and we rewrite the above equation for f as

f(x) = (1)/(2) \int _{R} \left( (x - t)/(+ |x - t|) (t)/(|t|) \right)d\mu(t).

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

\int _{|x|>1} (1)/(|x|) d\mu(x)< inf ty then the mapf(x) = (1)/(2) \int _{R ^{n}} \left( (x - y)/(+ |x - y|) (y)/(|y|) \right)d\mu(y) is quasisymmetric (in fact, it is δ-monotone for some δ depending on the measure μ).^[4]

Quasisymmetry and quasiconformality in Euclidean space

Let

\Omega and \Omega' 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, 2 r) is contained in \Omega , then f is η-quasisymmetric on B(x, 2 r) , where η depends only on K .

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, d_X) and (Y, d_Y) be two metric spaces and let η : [0,&nbsp;∞)&nbsp;→&nbsp;[0,&nbsp;∞) 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

(d _{Y(f(x),f(z))d} _{Y(f(y),f(t))})/(d _{Y(f(x),f(y))d} _{Y(f(z),f(t))}) \leq η \left( (d _{X(x,z)d} _{X(y,t)})/(d _{X(x,y)d} _{X(z,t)}) \right).

See also

• Douady–Earle extension

Notes and References

1. Book: Heinonen, Juha . Lectures on Analysis on Metric Spaces . Universitext . Springer-Verlag . New York . 2001 . x+140 . 978-0-387-95104-1.
2. Kovalev . Leonid V. . Quasiconformal geometry of monotone mappings . Journal of the London Mathematical Society . 75 . 2007 . 2 . 391–408 . 10.1112/jlms/jdm008. 10.1.1.194.2458 .
3. Beurling . A. . Ahlfors, L. . Lars Ahlfors . The boundary correspondence under quasiconformal mappings . Acta Math. . 96 . 1956 . 125–142. 10.1007/bf02392360. free .
4. Kovalev . Leonid . Maldonado . Diego . Wu . Jang-Mei . Jang-Mei Wu . Doubling measures, monotonicity, and quasiconformality . Math. Z. . 257 . 2007 . 3 . 525–545 . math/0611110 . 10.1007/s00209-007-0132-5. 119716883 .
5. Book: Buyalo. Sergei . Schroeder . Viktor. Elements of Asymptotic Geometry . EMS Monographs in Mathematics . American Mathematical Society . 2007 . 209 . 978-3-03719-036-4.