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.

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

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.

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.

...
Wikipedia