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 $f:X\to Y$ is said to be η-quasisymmetric for some increasing function $\eta :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ if for any triple $x,y,z$ of distinct points in $X$, we have:

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

Basic properties

Inverses are quasisymmetric

If $f:X\to Y$ is an invertible η-quasisymmetric map as above, then its inverse map is ${\eta }^{\prime }$-quasisymmetric, where ${\eta }^{\prime }(t)=1/{\eta }^{-1}(1/t).$

Quasisymmetric maps preserve relative sizes of sets

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

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

Examples

Weakly quasisymmetric maps

A map $f:X\to Y$ is said to be H-weakly-quasisymmetric for some $H>0$ if for all triples of distinct points $x,y,z$ in $X$, then

$|f(x)-f(y)|\le H|f(x)-f(z)|\text{ whenever }|x-y|\le |x-z|$

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 $f:H\to H$ on a Hilbert space $H$ is δ-monotone if for all $x,y$ in $H$,

$⟨f(x)-f(y),x-y⟩\ge \delta |f(x)-f(y)|\cdot |x-y|.$

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 $L:ℝ\to ℝ$.

Doubling measures

The real line

Quasisymmetric homeomorphisms of the real line to itself can be characterized in terms of their derivatives. An increasing homeomorphism $f:ℝ\to ℝ$ 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

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

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 ${ℝ}^n$: if μ is a doubling measure on ${ℝ}^n$ and

${\int }_{|x|>1}\frac{1}{|x|}d\mu (x)<\mathrm{\infty }$

then the map

$f(x)=\frac{1}{2}{\int }_{{ℝ}^n}(\frac{x-y}{|x-y|}+\frac{y}{|y|})d\mu (y)$

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

Quasisymmetry and quasiconformality in Euclidean space

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

Conversely, if $f:\mathrm{\Omega }\to {\mathrm{\Omega }}^{\prime }$ is K-quasiconformal and $B(x,2r)$ is contained in $\mathrm{\Omega }$, 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 $\eta :[0,+\mathrm{\infty })\to [0,+\mathrm{\infty })$ be an increasing function. An η-quasi-Möbius homeomorphism $f:X\to Y$ is a homeomorphism such that for every quadruple $x,y,z,t$ of distinct points in $X$, we have

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

See also

• Douady–Earle extension