In this paper, we prove a mean ergodic theorem for nonexpansive mappings in Hadamard (nonpositive curvature complete metric) spaces, which extends the Baillon nonlinear ergodic theorem. The main result shows that the sequence given by the Karcher means of iterations of a nonexpansive mapping with a nonempty fixed point set converges weakly to a fixed point of the mapping. This result also remains true for a 1-parameter continuous semigroup of contractions.

中文翻译：

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Hadamard空间中非扩张映射的平均遍历定理

在本文中，我们证明了Hadamard（非正曲率完全度量）空间中非膨胀映射的平均遍历定理，它扩展了Baillon非线性遍历定理。主要结果表明，具有非空固定点集的非扩展映射的Karcher迭代方式给出的序列弱收敛至映射的固定点。对于收缩的1参数连续半群，该结果也适用。