A metric fixed point theorem and some of its applications

TL;DR

This paper introduces a new fixed point theorem for isometries in metric spaces with applications to Banach spaces, CAT(0)-spaces, and ergodic theory, extending classical results and providing new insights into invariant functionals.

Contribution

It presents a novel fixed point theorem applicable to a broad class of spaces, including non-locally compact and non-proper spaces, with implications for ergodic theory and invariant subspace problems.

Findings

Proves a general fixed point theorem for isometries with metric functionals.

Derives a new mean ergodic theorem generalizing von Neumann's theorem.

Shows every bounded invertible operator admits a nontrivial invariant metric functional.

Abstract

A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new even for isometries of Banach spaces as well as for non-locally compact CAT(0)-spaces and injective spaces. Examples of actions on non-proper CAT(0)-spaces come from the study of diffeomorphism groups, birational transformations, and compact K\"ahler manifolds. A special case of the fixed point theorem provides a novel mean ergodic theorem that in the Hilbert space case implies von Neumann's theorem. The theorem accommodates classically fixed-point-free isometric maps such as those of Kakutani, Edelstein, Alspach and Prus. Moreover, from the main theorem together with some geometric arguments of independent interest, one can deduce that every bounded invertible operator of a Hilbert space admits a nontrivial invariant metric functional on the space of positive operators. This is a result in the direction of the invariant subspace problem although its full meaning is dependent on a future determination of such metric functionals.