The main purpose of this paper is to prove the mean ergodic theorem for nonexpansive mappings and semigroups in locally compact Hadamard spaces, including finite dimensional Hadamard manifolds. The main tool for proving ergodic convergence is the almost periodicity of orbits of a nonexpansive mapping. Therefore, in the first part of the paper, we study almost periodicity (and as a special case, periodicity) in metric and Hadamard spaces. Then, we prove a mean ergodic theorem for nonexpansive mappings and continuous semigroups of contractions in locally compact Hadamard spaces. Finally, an application to the asymptotic behavior of the first order evolution equation associated to the monotone vector field on Hadamard manifolds is presented.

中文翻译：

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Hadamard 空间中非扩张映射和半群的几乎周期性和遍历定理

本文的主要目的是证明局部紧哈达玛空间（包括有限维哈达玛流形）中非膨胀映射和半群的平均遍历定理。证明遍历收敛的主要工具是非膨胀映射的轨道几乎具有周期性。因此，在论文的第一部分，我们研究了度量空间和哈达玛空间中的几乎周期性（并且作为一种特殊情况，周期性）。然后，我们证明了局部紧阿达玛空间中非膨胀映射和收缩的连续半群的平均遍历定理。最后，介绍了与 Hadamard 流形上的单调矢量场相关的一阶演化方程的渐近行为的应用。