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, dX) and (Y, dY) 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

d Y ( f ( x ) , f ( y ) ) d Y ( f ( x ) , f ( z ) ) ≤ η ( d X ( x , y ) d X ( x , z ) ) . {\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 η ′ {\displaystyle \eta '}-quasisymmetric, where η ′ ( t ) = 1 / η − 1 ( 1 / t ) . {\textstyle \eta '(t)=1/\eta ^{-1}(1/t).}

Quasisymmetric maps preserve relative sizes of sets

If A {\displaystyle A} and B {\displaystyle B} are subsets of X {\displaystyle X} and A {\displaystyle A} is a subset of B {\displaystyle B}, then η − 1 ( diam ⁡ B diam ⁡ A ) 2 ≤ diam ⁡ f ( B ) diam ⁡ f ( A ) ≤ 2 η ( diam ⁡ B diam ⁡ A ) . {\displaystyle {\frac {\eta ^{-1}({\frac {\operatorname {diam} B}{\operatorname {diam} A}})}{2}}\leq {\frac {\operatorname {diam} f(B)}{\operatorname {diam} f(A)}}\leq 2\eta \left({\frac {\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 H > 0 {\displaystyle H>0} if for all triples of distinct points x , y , z {\displaystyle x,y,z} in X {\displaystyle X}, then

| f ( x ) − f ( y ) | ≤ H | f ( x ) − f ( z ) | whenever | x − y | ≤ | x − z | {\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 X {\displaystyle X} is connected and X {\displaystyle X} and Y {\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,

⟨ f ( x ) − f ( y ) , x − y ⟩ ≥ δ | f ( x ) − f ( y ) | ⋅ | x − y | . {\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.

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 + ∫ 0 x d μ ( t ) . {\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

f ( x ) = 1 2 ∫ R ( x − t | x − t | + t | t | ) d μ ( t ) . {\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

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

then the map

f ( x ) = 1 2 ∫ R n ( x − y | x − y | + y | y | ) d μ ( y ) {\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).

Quasisymmetry and quasiconformality in Euclidean space

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

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

Definition

Let (X, dX) and (Y, dY) 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

d Y ( f ( x ) , f ( z ) ) d Y ( f ( y ) , f ( t ) ) d Y ( f ( x ) , f ( y ) ) d Y ( f ( z ) , f ( t ) ) ≤ η ( d X ( x , z ) d X ( y , t ) d X ( x , y ) d X ( z , t ) ) . {\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