Quasisymmetric embeddings of metric spaces

or all such triples a, b, x.We also say that / is 4-quasisymmetric if (l .2) is satisfied with a given function 4. Otherwise stated, an embedding f: X-I is 4-QS if and only if la-xl=tlb-xl implies lf@)-f(x)l=aQ)lf(b)-f(x)l for all a,b,x(X and l>0.If/is a-QS with 4(l):1, we say that/is a similarity.Thus/is a similarity if and only if there is Z>0 such that lf@)-fQ)l:Llx-yl for all x, y(X.A curious fact is that for a large class of spaces X, including all connected spaces, every QS embedding f: X*I is 4-QS with 4 of the form ry(r):Cmax(t",tttn1 (Corollary 3,12).The term "quasisymmetric" has been used in the theory of quasiconformal maps LV, II.7.1 for increasing embeddings f: /*Rt, where / cRl is an interval, such that for some constant fI, lf @)-f (x)l = H lf (b)-f (x)l whenever la-xl=lb-xl.The concept was introduced by Beurling and AhlforsBA, the term by Kelingos Ks.For general metric spaces, we shall call an embedding