An L 1 Ergodic Theorem with Values in a Non-Positively Curved Space Via a Canonical Barycenter Map

We give a general version of the Birkhoff ergodic theorem for functions taking values in non-positively curved spaces. In this setting, the notion of a Birkhoff sum is replaced by that of a barycenter along the orbits. The construction of an appropriate barycenter map is the core of this note. As a byproduct of our construction, we prove a fixed point theorem for actions by isometries on a Buseman space. © 2012 Cambridge University Press.