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 ) {\displaystyle (X,d_{X})} and ( Y , d Y ) {\displaystyle (Y,d_{Y})} be two metric spaces. A homeomorphism f : X → Y {\displaystyle f\colon X\to Y} is said to be η-quasisymmetric for some increasing function η : [ 0 , + ∞ ) → [ 0 , + ∞ ) {\displaystyle \eta :[0,+\infty )\to [0,+\infty )} if for any triple x , y , z {\displaystyle x,y,z} of distinct points in X {\displaystyle 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 {\displaystyle f\colon X\to 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
1 2 η − 1 ( diam B diam A ) ≤ diam f ( B ) diam f ( A ) ≤ 2 η ( diam B diam A ) . {\displaystyle {\frac {1}{2}}\eta ^{-1}\left({\frac {\operatorname {diam} B}{\operatorname {diam} A}}\right)\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 {\displaystyle f\colon X\to 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
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