Antezana, Jorge; Ghiglioni, Eduardo; Stojanoff, Demetrio Ergodic theorem in CAT(0) spaces in terms of inductive means. (English) Zbl 1517.37009 Ergodic Theory Dyn. Syst. 43, No. 5, 1433-1454 (2023). Summary: Let \((G,+)\) be a compact, abelian, and metrizable topological group. In this group we take \(g\in G\) such that the corresponding automorphism \(\tau_g\) is ergodic. The main result of this paper is a new ergodic theorem for functions in \(L^1(G,M)\), where \(M\) is a Hadamard space. The novelty of our result is that we use inductive means to average the elements of the orbit \(\{\tau_g^n(h)\}_{n\in\mathbb{N}}\). The advantage of inductive means is that they can be explicitly computed in many important examples. The proof of the ergodic theorem is done firstly for continuous functions, and then it is extended to \(L^1\) functions. The extension is based on a new construction of mollifiers in Hadamard spaces. This construction has the advantage that it only uses the metric structure and the existence of barycenters, and does not require the existence of an underlying vector space. For this reason, it can be used in any Hadamard space, in contrast to those results that need to use the tangent space or some chart to define the mollifier. MSC: 37A30 Ergodic theorems, spectral theory, Markov operators 22D40 Ergodic theory on groups 22F05 General theory of group and pseudogroup actions Keywords:ergodic theorem; inductive means; barycenter; Hadamard space; non-positively curved space; mollifier Software:Matrix Means Toolbox × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Alexandrov, A. D.. A theorem on triangles in a metric space and some applications. Tr. Mat. Inst. Steklova38 (1951), 5-23. · Zbl 0049.39501 [2] Austin, T.. A CAT(0) valued pointwise ergodic theorem. J. Topol. Anal.3 (2011), 145-152. · Zbl 1221.37006 [3] Bačák, M.. 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