Non-linear ergodic theorems in complete non-positive curvature metric spaces

Abstract

Hadamard (or complete $CAT(0)$) spaces are complete, non-positive curvature, metric spaces. Here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. Our results extend the standard non-linear ergodic theorems for non-expansive maps on real Hilbert spaces, to non-expansive maps on Hadamard spaces, which include for example (possibly infinite-dimensional) complete simply connected Riemannian manifolds with non-positive sectional curvature.