quasisymmetric map

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

:$\backslash frac\{d\_Y(f(x),f(y))\}\{d\_\{Y\}(f(x),f(z))\}\; \backslash leq\; \backslash eta\backslash left(\backslash frac\{d\_X(x,y)\}\{d\_X(x,z)\}\backslash right).$

; Inverses are quasisymmetric : If f : X → Y is an invertible η-quasisymmetric map as above, then its inverse map is $\backslash eta\text{'}$-quasisymmetric, where $\backslash eta\text{'}(t)\; =\; 1/\backslash eta^\{-1\}(1/t).$

; Quasisymmetric maps preserve relative sizes of sets : If $A$ and $B$ are subsets of $X$ and $A$ is a subset of $B$, then

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

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

:$|f(x)-f(y)|\backslash leq\; H|f(x)-f(z)|\backslash ;\backslash ;\backslash ;\backslash text\{\; whenever\; \}\backslash ;\backslash ;\backslash ;\; |x-y|\backslash leq\; |x-z|$

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.

A monotone map f:H → H on a Hilbert space H is δ-monotone if for all x and y in H,

:$\backslash langle\; f(x)-f(y),x-y\backslash rangle\backslash geq\; \backslash delta\; |f(x)-f(y)|\backslash cdot|x-y|.$

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.{{cite journal | last = Kovalev | first = Leonid V. | title= Quasiconformal geometry of monotone mappings | journal = Journal of the London Mathematical Society | volume = 75 | year = 2007 | number = 2 | pages = 391–408 | doi=10.1112/jlms/jdm008| citeseerx = 10.1.1.194.2458 }}

Quasisymmetric homeomorphisms of the real line to itself can be characterized in terms of their derivatives.{{cite journal| last = Beurling | first = A. |author2=Ahlfors, L. |authorlink2=Lars Ahlfors | title = The boundary correspondence under quasiconformal mappings | journal = Acta Math. | volume= 96 | year = 1956 | pages = 125–142| doi=10.1007/bf02392360| doi-access = free }} 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

:$f(x)=C+\backslash int\_0^x\; \backslash ,\; d\backslash mu(t).$

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

:$f(x)\; =\; \backslash frac\{1\}\{2\}\backslash int\_\{\backslash mathbb\{R\}\}\backslash left(\backslash frac\{x-t\}$

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

:$\backslash int\_$

then the map

:$f(x)\; =\; \backslash frac\{1\}\{2\}\backslash int\_\{\backslash mathbb\{R\}^\{n\}\}\backslash left(\backslash frac\{x-y\}$

is quasisymmetric (in fact, it is δ-monotone for some δ depending on the measure μ).{{cite journal| last1 = Kovalev | first1 = Leonid | last2 = Maldonado | first2 = Diego | last3 = Wu | first3 = Jang-Mei | author3-link= Jang-Mei Wu | title = Doubling measures, monotonicity, and quasiconformality |journal= Math. Z. | volume = 257 | year = 2007 | number = 3 | pages = 525–545 | arxiv=math/0611110 | doi=10.1007/s00209-007-0132-5| s2cid = 119716883 }}

Let $\backslash Omega$ and $\backslash Omega\text{'}$ 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 $\backslash eta$.

Conversely, if f : Ω → Ω´ is K-quasiconformal and $B(x,2r)$ is contained in $\backslash Omega$, then $f$ is η-quasisymmetric on $B(x,2r)$, where $\backslash eta$ depends only on $K$.

A related but weaker condition is the notion of quasi-Möbius maps where instead of the ratio only the cross-ratio is considered:{{cite book| last1 = Buyalo| first1 = Sergei | last2 = Schroeder | first2 = Viktor| title = Elements of Asymptotic Geometry | series = EMS Monographs in Mathematics | publisher = American Mathematical Society | year = 2007 | pages = 209 | isbn = 978-3-03719-036-4}}

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

:$\backslash frac\{d\_Y(f(x),f(z))d\_Y(f(y),f(t))\}\{d\_Y(f(x),f(y))d\_Y(f(z),f(t))\}\; \backslash leq\; \backslash eta\backslash left(\backslash frac\{d\_X(x,z)d\_X(y,t)\}\{d\_X(x,y)d\_X(z,t)\}\backslash right).$

• Douady–Earle extension

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