Quasisymmetric map

For quasisymmetric functions in algebraic combinatorics, see quasisymmetric function.

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

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

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 ${\textstyle \eta '(t)=1/\eta ^{-1}(1/t).}$

Quasisymmetric maps preserve relative sizes of sets

If $A$ and $B$ are subsets of $X$ and $A$ is a subset of $B$, then

${\frac {\eta ^{-1}({\frac {\operatorname {diam} B}{\operatorname {diam} A}})}{2}}\leq {\frac {\operatorname {diam} f(B)}{\operatorname {diam} f(A)}}\leq 2\eta \left({\frac {\operatorname {diam} B}{\operatorname {diam} A}}\right).$

Examples

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.

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

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

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, 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

${\frac {d_{Y}(f(x),f(z))d_{Y}(f(y),f(t))}{d_{Y}(f(x),f(y))d_{Y}(f(z),f(t))}}\leq \eta \left({\frac {d_{X}(x,z)d_{X}(y,t)}{d_{X}(x,y)d_{X}(z,t)}}\right).$

See also

• Douady–Earle extension