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.^[1]

Definition

Let $(X,d_{X})$ and $(Y,d_{Y})$ be two metric spaces. A homeomorphism $f\colon X\to Y$ is said to be η-quasisymmetric for some increasing function ${\displaystyle \eta$ if 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\colon X\to 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 {1}{2}}\eta ^{-1}\left({\frac {\operatorname {diam} B}{\operatorname {diam} A}}\right)\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 $f\colon X\to Y$ 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 $f\colon H\to H$ on a Hilbert space $H$ is δ-monotone if for all $x,y$ 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 $\arccos(\delta )<\pi /2$ .

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 $L:\mathbb {R} \rightarrow \mathbb {R}$ .^[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 $f:\mathbb {R} \rightarrow \mathbb {R}$ 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 $\mathbb {R} ^{n}$ : if μ is a doubling measure on $\mathbb {R} ^{n}$ 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).^[4]

Quasisymmetry and quasiconformality in Euclidean space

Let $\Omega$ and $\Omega '$ be open subsets of $\mathbb {R} ^{n}$ . If $f\colon \Omega \to \Omega '$ is η-quasisymmetric, then it is also K-quasiconformal, where $K>0$ is a constant depending on $\eta$ .

Conversely, if $f\colon \Omega \to \Omega '$ 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:^[5]

Definition

Let $(X,d_{X})$ and $(Y,d_{Y})$ be two metric spaces and let ${\displaystyle \eta$ be an increasing function. An η-quasi-Möbius homeomorphism $f\colon X\to Y$ is a homeomorphism such that 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).

Quasisymmetric map

Definition

Basic properties

Examples

Weakly quasisymmetric maps

δ-monotone maps

Doubling measures

The real line

Euclidean space

Quasisymmetry and quasiconformality in Euclidean space

Quasi-Möbius maps

Definition

See also

References

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