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, 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, ∞) → [0, ∞) such that for any triple x, y, z of distinct points in X, we have

[math]\displaystyle{ \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 : X → Y is an invertible η-quasisymmetric map as above, then its inverse map is [math]\displaystyle{ \eta' }[/math]-quasisymmetric, where [math]\displaystyle{ \eta'(t) = 1/\eta^{-1}(1/t). }[/math]

Quasisymmetric maps preserve relative sizes of sets

If [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] are subsets of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ A }[/math] is a subset of [math]\displaystyle{ B }[/math], then

[math]\displaystyle{ \frac{1}{2\eta(\frac{\operatorname{diam}A}{\operatorname{diam}B})}\leq \frac{\operatorname{diam}f(B)}{\operatorname{diam}f(A)}\leq \eta\left(\frac{2\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]\displaystyle{ H\gt 0 }[/math] if for all triples of distinct points [math]\displaystyle{ x,y,z }[/math] in [math]\displaystyle{ X }[/math], then

[math]\displaystyle{ |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]\displaystyle{ X }[/math] is connected and [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] 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,

[math]\displaystyle{ \langle f(x)-f(y),x-y\rangle\geq \delta |f(x)-f(y)|\cdot|x-y|. }[/math]

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.

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

[math]\displaystyle{ f(x)=C+\int_0^x \, d\mu(t). }[/math]

Euclidean space

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

[math]\displaystyle{ f(x) = \frac{1}{2}\int_{\mathbb{R}}\left(\frac{x-t}{|x-t|}+\frac{t}{|t|}\right)d\mu(t). }[/math]

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

[math]\displaystyle{ \int_{|x|\gt 1}\frac{1}{|x|}\,d\mu(x)\lt \infty }[/math]

then the map

[math]\displaystyle{ f(x) = \frac{1}{2}\int_{\mathbb{R}^{n}}\left(\frac{x-y}{|x-y|}+\frac{y}{|y|}\right)\,d\mu(y) }[/math]

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

Quasisymmetry and quasiconformality in Euclidean space

Let [math]\displaystyle{ \Omega }[/math] and [math]\displaystyle{ \Omega' }[/math] be open subsets of ℝⁿ. If f : Ω → Ω´ is η-quasisymmetric, then it is also K-quasiconformal, where [math]\displaystyle{ K\gt 0 }[/math] is a constant depending on [math]\displaystyle{ \eta }[/math].

Conversely, if f : Ω → Ω´ is K-quasiconformal and [math]\displaystyle{ B(x,2r) }[/math] is contained in [math]\displaystyle{ \Omega }[/math], then [math]\displaystyle{ f }[/math] is η-quasisymmetric on [math]\displaystyle{ B(x,2r) }[/math], where [math]\displaystyle{ \eta }[/math] depends only on [math]\displaystyle{ K }[/math].

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, d_X) and (Y, d_Y) 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

[math]\displaystyle{ \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). }[/math]

See also

• Douady–Earle extension