Quasisymmetric map

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

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

Basic properties

Inverses are quasisymmetric 

If f : X → Y is an invertible η-quasisymmetric map as above, then its inverse map is η ′ -quasisymmetric, where η ′ (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 1 2 η ( diam  B diam  A ) ≤ diam  f ( B ) diam  f ( A ) ≤ η ( diam  B diam  A ) .

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 ) | ≤ H | f ( x ) − f ( z ) |  whenever  | x − y | ≤ | x − z |

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,

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

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 ) .

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 ) .

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 ) < ∞

then the map

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