Quasisymmetric map

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.

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


 * $$ \frac{d_Y(f(x),f(y))}{d_{Y}(f(x),f(z))} \leq \eta\left(\frac{d_X(x,y)}{d_X(x,z)}\right). $$

Basic properties

 * Inverses are quasisymmetric : If f : X → Y is an invertible η-quasisymmetric map as above, then its inverse map is $$\eta'$$-quasisymmetric, where $\eta'(t) = 1/\eta^{-1}(1/t).$
 * Quasisymmetric maps preserve relative sizes of sets : If $$A$$ and $$B$$ are subsets of $$X$$ and $$A$$ is a subset of $$B$$, then
 * $$ \frac{\eta^{-1}(\frac{\operatorname{diam} B}{\operatorname{diam} A})}{2}\leq \frac{\operatorname{diam}f(B)}{\operatorname{diam}f(A)}\leq 2\eta\left(\frac{\operatorname{diam} B}{\operatorname{diam}A}\right).$$

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 $$X$$, then
 * $$ |f(x)-f(y)|\leq H|f(x)-f(z)|\;\;\;\text{ whenever }\;\;\; |x-y|\leq |x-z|$$

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.

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,
 * $$ \langle f(x)-f(y),x-y\rangle\geq \delta |f(x)-f(y)|\cdot|x-y|.$$

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.

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


 * $$ f(x)=C+\int_0^x \, d\mu(t).$$

Euclidean space
An analogous result holds in Euclidean space. Suppose C = 0 and we rewrite the above equation for f as
 * $$ f(x) = \frac{1}{2}\int_{\mathbb{R}}\left(\frac{x-t}{|x-t|}+\frac{t}{|t|}\right)d\mu(t).$$

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
 * $$ \int_{|x|>1}\frac{1}{|x|}\,d\mu(x)<\infty$$

then the map
 * $$ f(x) = \frac{1}{2}\int_{\mathbb{R}^{n}}\left(\frac{x-y}{|x-y|}+\frac{y}{|y|}\right)\,d\mu(y)$$

is quasisymmetric (in fact, it is δ-monotone for some δ depending on the measure μ).

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 $$\eta$$.

Conversely, if f : Ω → Ω´ is K-quasiconformal and $$B(x,2r)$$ is contained in $$\Omega$$, then $$f$$ is η-quasisymmetric on $$B(x,2r)$$, where $$\eta$$ 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:

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


 * $$ \frac{d_Y(f(x),f(z))d_Y(f(y),f(t))}{d_Y(f(x),f(y))d_Y(f(z),f(t))} \leq \eta\left(\frac{d_X(x,z)d_X(y,t)}{d_X(x,y)d_X(z,t)}\right). $$