Metric spaces admitting only trivial weak contractions

Abstract

If $(X,d)$ is a metric space then the map $f\colon X\to X$ is defined to be a weak contraction if $d(f(x),f(y))<d(x,y)$ for all $x,y\in X$, $x\neq y$. We determine the simplest non-closed sets $X\subseteq \mathbb{R}^n$ in the sense of descriptive set theoretic complexity such that every weak contraction $f\colon X\to X$ is constant. In order to do so, we prove that there exists a non-closed $F_{\sigma}$ set $F\subseteq \mathbb{R}$ such that every weak contraction $f\colon F\to F$ is constant. Similarly, there exists a non-closed $G_{\delta}$ set $G\subseteq \mathbb{R}$ such that every weak contraction $f\colon G\to G$ is constant. These answer questions of M. Elekes. We use measure theoretic methods, first of all the concept of generalized Hausdorff measure.