Sufficient conditions for quasisymmetry of mappings of the real axis and the plane
Let \(X\) and \(Y\) be metric spaces, \(\eta:[0,+\infty) \to (0,+\infty)\) is a homeomorphism. A mapping \(f:X\to Y\) is called \(\eta\)-quasisymmetric if, for all \(x,y,z\in X\), \[ |f(x)-f(z)|/ |f(y)-f(z)|\leq \eta(|x-z|/|y-z|). \] The authors construct an example of a homeomorphic embedding \(f\:\mathbb R^1 \to \mathbb R^3\) which satisfies an analog of the Ahlfors \(M\)-condition but is not quasisymmetric. The authors say that a topological embedding \(f:I\to X\), \(I=(a,b)\subset \mathbb R^1\) satisfies the midpoint condition with constant \(H\) if \( |f((x+y)/2)-f(x)|\leq H |f(y)-f(x)|\) for all \(x,y\in I\). It is shown that this condition with constant \(H<1\) is sufficient for quasisymmetry of the mapping \(f:I\to \mathbb R^n\). Some other sufficient conditions for quasisymmetry of mappings in the plane are established.
scientific article; zbMATH DE number 1432315