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

Document Type : Research Paper

Authors

Tarbiat Modares University

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.

Keywords

### History

• Receive Date: 08 November 2009
• Revise Date: 09 March 2012
• Accept Date: 05 January 2010
• First Publish Date: 15 September 2011