Anders Karlsson - A fixed point theorem for isometries of metric spaces

This talk was part of the Thematic Programme on "Geometry beyond Riemann: Curvature and Rigidity" held at the ESI September 11 — November 3, 2023.

A new general fixed point theorem for isometric maps of metric spaces which admits a weak convex bicombing that can be thought of as Busemann nonpositive curvature for a distinguished class of geodesics, will be explained. The spaces include all Banach spaces, CAT(0)-spaces, injective metric spaces as well as the space Pos of positive operators equipped with Thompson’s metric and certain spaces of Riemannian or Kähler metrics on which diffeomorphisms act by isometry. The fixed point is given as a metric functional, which is an extension of the concept of Busemann functions, thus providing a fixed point even if there is none in the classical sense. One particular application of the fixed point theorem is a new mean ergodic theorem valid in all Banach spaces generalizing von Neumann’s theorem in case of Hilbert spaces. Another application provides a non-trivial invariant metric functional on Pos to any invertible bounded linear operator of a Hilbert space.