Quasisymmetric map

For quasisymmetric functions in algebraic combinatorics, see quasisymmetric function.

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

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

Basic properties

Inverses are quasisymmetric

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

Quasisymmetric maps preserve relative sizes of sets

If ${\displaystyle A}$ and ${\displaystyle B}$ are subsets of ${\displaystyle X}$ and ${\displaystyle A}$ is a subset of ${\displaystyle B}$, then

${\displaystyle {\frac {1}{2\eta ({\frac {\operatorname {diam} A}{\operatorname {diam} B}})}}\leq {\frac {\operatorname {diam} f(B)}{\operatorname {diam} f(A)}}\leq \eta \left({\frac {2\operatorname {diam} B}{\operatorname {diam} A}}\right).}$

Examples

Weakly quasisymmetric maps

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.

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,

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

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]

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

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

Euclidean space

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]

Quasisymmetry and quasiconformality in Euclidean space

Let ${\displaystyle \Omega }$ and ${\displaystyle \Omega '}$ be open subsets of ℝⁿ. If f : Ω → Ω´ is η-quasisymmetric, then it is also K-quasiconformal, where ${\displaystyle K>0}$ is a constant depending on ${\displaystyle \eta }$.

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

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: ^[5]

Definition

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

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

See also

• Douady–Earle extension

References

1. Heinonen, Juha (2001). Lectures on Analysis on Metric Spaces. Universitext. New York: Springer-Verlag. pp. x+140. ISBN   978-0-387-95104-1.
2. 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.
3. Beurling, A.; Ahlfors, L. (1956). "The boundary correspondence under quasiconformal mappings". Acta Math. 96: 125–142. doi: 10.1007/bf02392360 .
4. 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. S2CID   119716883.
5. Buyalo, Sergei; Schroeder, Viktor (2007). Elements of Asymptotic Geometry. EMS Monographs in Mathematics. American Mathematical Society. p. 209. ISBN   978-3-03719-036-4.