Quasisymmetric mappings in b-metric spaces
Let \((X,d)\) and \((Y,\rho)\) be \(b\)-metric spaces with the coefficients \(K_1\) and \(K_2\), respectively and let \(f:X\to Y\) be an \(\eta\)-quasisymmetric embedding (i.e., \(\eta:\mathbb R^+\to \mathbb R^+\) is a homeomorphism so that \(d(x,a)\le td(x,b)\) implies \(\rho(f(x),f(a))\le\eta(t)\rho(f(x),f(b))\) for all triples \(a,b,x\in X\) and for every \(t\in \mathbb R^+\)). The authors prove that \(f\) maps bounded subspaces to bounded subspaces and, if \(A\subseteq B\subseteq X\), \(0<\operatorname{diam} A,\operatorname{diam} B<\infty\), then \(\operatorname{diam} f(B)\) is finite and the following holds: \[ \frac1{2K_2\eta(\frac{\operatorname{diam} B}{\operatorname{diam} A})}\le\frac{\operatorname{diam} f(A)}{\operatorname{diam} f(B)}\le\eta\Bigl(\frac{2K_1\operatorname{diam} A}{\operatorname{diam} B}\Bigr), \] thus adapting a result known for metric spaces. They also describe \(\eta\)-quasisymmetric embeddings \(f\) for which the image \(f(X)\) of a \(b\)-metric space \(X\) with coefficient \(K_1\) is again a \(b\)-metric space with coefficient \(K_2\). Finally, a similar question is investigated for additive metric spaces.
scientific article; zbMATH DE number 7378023
scientific article