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'' :  f(x)-f(y), \leq H, f(x)-f(z), \;\;\;\text\;\;\; , x-y, \leq , x-z,

Not all weakly quasisymmetric maps are quasisymmetric. However, if $X$ is connected and $X$ and $Y$ are doubling measures and metric spaces, 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#Monotonicity in functional analysis, 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
:$f(x) = \frac\int_\left(\frac+\frac\right)d\mu(t).$

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 ℝ^''n'': if ''μ'' is a doubling measure on ℝ^''n'' and
:$\int_\frac\,d\mu(x)<\infty$
then the map
:$f(x) = \frac\int_\left(\frac+\frac\right)\,d\mu(y)$
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 ℝ^''n''. 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

:$\frac \leq \eta\left(\frac\right).$

See also

*Douady–Earle extension

References

{{reflist

Homeomorphisms
Geometry
Mathematical analysis
Metric geometry