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Almost periodicity and ergodic theorems for nonexpansive mappings and semigroups in Hadamard spaces

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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 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.

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The authors are grateful to the referees for their careful reading and valuable comments and suggestions.

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Correspondence to Hadi Khatibzadeh.

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Communicated by Anthony To-Ming Lau.

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Khatibzadeh, H., Pouladi, H. Almost periodicity and ergodic theorems for nonexpansive mappings and semigroups in Hadamard spaces. Semigroup Forum 101, 716–733 (2020).

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