Quasisymmetric map

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

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

Not all weakly quasisymmetric maps are quasisymmetric. However, if ${\displaystyle X}$ is connected and ${\displaystyle X}$ and ${\displaystyle Y}$ 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,

${\displaystyle \langle f(x)-f(y),x-y\rangle \geq \delta |f(x)-f(y)|\cdot |x-y|.}$

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.[2]

Quasisymmetric homeomorphisms of the real line to itself can be characterized in terms of their derivatives.[3] 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

${\displaystyle f(x)=C+\int _{0}^{x}\,d\mu (t).}$

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

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

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 ℝⁿ: if μ is a doubling measure on ℝⁿ and

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

then the map

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

is quasisymmetric (in fact, it is δ-monotone for some δ depending on the measure μ).[4]

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

${\displaystyle {\frac {d_{Y}(f(x),f(z))d_{Y}(f(y),f(t))}{d_{Y}(f(x),f(y))d_{Y}(f(z),f(t))}}\leq \eta \left({\frac {d_{X}(x,z)d_{X}(y,t)}{d_{X}(x,y)d_{X}(z,t)}}\right).}$