On distance dependent maps on the circle and on the real line (Q1811231)

A map \(f\) from a metric space \((X,d)\) into itself is called {distance compatible} if \(d(f(x),f(y))\) depends only on \(d(x,y)\), for any two points \(x,y\) of \(X\). The authors prove that a map \(f:\mathbb{R\rightarrow R}\) is distance compatible if and only if \(g:=f-f(0)\) satisfies the functional equation \(g(x+y)=g(x)+g(y)\). This implies that the only Lebesgue measurable distance compatible maps on \(\mathbb{R}\) are those of the form \(x\mapsto mx+b\) with \(m,b\in \mathbb{R}\). By a similar reasoning, it turns out that on the circle \(S^{1}:=\{z\in \mathbb{C}:\left| z\right| =1\}\) the only such maps are those of the form \(z\mapsto cz^{n}\) with \(c\in S^{1}\) and \(n\in \mathbb{Z}\).