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28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. 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

$\frac{d_Y(f(x),f(y))}{d_Y(f(x),f(z))}\le \eta \left(\frac{d_X(x,y)}{d_X(x,z)}\right).$

Basic properties

Inverses are quasisymmetric

If f : X → Y is an invertible η-quasisymmetric map as above, then its inverse map is ή-quasisymmetric, where ή(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

$\frac{1}{2\eta (\frac{\text{diam }B}{\text{diam }A})}\le \frac{\text{diam }f(B)}{\text{diam }f(A)}\le \eta \left(\frac{\text{diam }B}{\text{diam }A}\right).$

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

$|f(x)-f(y)|\le H|f(x)-f(z)|\text{ whenever }|x-y|\le |x-z|$

Not all weakly quasisymmetric maps are quasisymmetric. However, if X is connected and 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 we have the luck of knowing the two 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,

$⟨f(x)-f(y),x-y⟩\ge \delta |f(x)-f(y)|\cdot |x-y|.$

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.

Quasisymmetric maps and 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

$f(x)=C+\underset{0}{\overset{x}{\int }}d\mu (t).$

Euclidean space

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

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

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

${\int }_{\left|x\right|>1}\frac{1}{\left|x\right|}d\mu (x)<\mathrm{\infty }$

then the map

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

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 ℝⁿ. 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, 2r) is contained in Ω, then f is η-quasisymmetric on B(x, r), where η depends only on K.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.