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Let (G,+) be a compact, abelian, and metrizable topological group. In thisgroup we take $g\in G$ such that the corresponding automorphism t_g is ergodic.The main result of this paper is a new ergodic theorem for functions inL^1(G,M), where M is a Hadamard space. The novelty of our result is that we useinductive means to average the elements of the orbit {t_g^n(h)}_{n\in N}. Theadvantage of inductive means is that they can be explicitly computed in manyimportant examples. The proof of the ergodic theorem is done firstly forcontinuous functions, and then it is extended to L^1 functions. The extensionis based in a new construction of mollifiers in Hadamard spaces. Thisconstruction has the advantage that it only uses the metric structure and theexistence of barycenters, and do not require the existence of an underlyingvector space. For this reason, it can be used in any Hadamard space, incontrast with those results that need to use the tangent space or some chart todefine the mollifier.

Ergodic theorem in Hadamard spaces in terms of inductive means