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" : [0, ∞) → [0, ∞) such that for any triple "x", "y", "z" of distinct points in "X", we have
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", we have
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",
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
Euclidean space.
An analogous result holds in Euclidean space. Suppose "C" = 0 and we rewrite the above equation for "f" as 
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
is quasisymmetric (in fact, it is "δ"-monotone for some "δ" depending on the measure "μ").
Quasisymmetry and quasiconformality in Euclidean space.
Let "Ω" and "Ω´" 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 "η".
Conversely, if "f" : Ω → Ω´ is "K"-quasiconformal and "B"("x", 2"r") is contained in "Ω", then "f" is "η"-quasisymmetric on "B"("x", "r"), where "η" 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. A homeomorphism "f":"X" → "Y" is said to be η-quasi-Möbius if there is an increasing function "η" : [0, ∞) → [0, ∞) such that for any quadrupple "x", "y", "z", "t" of 
distinct points in "X", we have