Metric spaces admitting only trivial weak contractions

Richárd Balka¹

¹ Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences PO Box 127, 1364 Budapest, Hungary

Fundamenta Mathematicae, Tome 221 (2013) no. 1, pp. 83-94

Résumé

If $(X,d)$ is a metric space then a 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.

Détail

DOI : 10.4064/fm221-1-4

Keywords: metric space map colon defined weak contraction neq determine simplest non closed sets subseteq mathbb sense descriptive set theoretic complexity every weak contraction colon constant order prove there exists non closed sigma set subseteq mathbb every weak contraction colon constant similarly there exists non closed delta set subseteq mathbb every weak contraction colon constant these answer questions elekes measure theoretic methods first concept generalized hausdorff measure