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.
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