# 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.[1]

## 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

## Basic properties

- Inverses are quasisymmetric
- If
*f*:*X*→*Y*is an invertible*η*-quasisymmetric map as above, then its inverse map is -quasisymmetric, where (*t*) = 1/*η*(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

## 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.[2]

## Doubling measures

### The real line

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

### 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

then the map

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

## 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*.

## See also

## References

- Heinonen, Juha (2001).
*Lectures on Analysis on Metric Spaces*. Universitext. New York: Springer-Verlag. pp. x+140. ISBN 978-0-387-95104-1. - Kovalev, Leonid V. (2007). "Quasiconformal geometry of monotone mappings".
*Journal of the London Mathematical Society*.**75**(2): 391–408. CiteSeerX 10.1.1.194.2458. doi:10.1112/jlms/jdm008. - Beurling, A.; Ahlfors, L. (1956). "The boundary correspondence under quasiconformal mappings".
*Acta Math*.**96**: 125–142. doi:10.1007/bf02392360. - Kovalev, Leonid; Maldonado, Diego; Wu, Jang-Mei (2007). "Doubling measures, monotonicity, and quasiconformality".
*Math. Z*.**257**(3): 525–545. arXiv:math/0611110. doi:10.1007/s00209-007-0132-5.