Ergodic theorem in Hadamard spaces in terms of inductive means

Abstract

Let (G,+) be a compact, abelian, and metrizable topological group. In this group 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 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 {t_g^n(h)}_{n\in 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 in 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 do not require the existence of an underlying vector space. For this reason, it can be used in any Hadamard space, in contrast with those results that need to use the tangent space or some chart to define the mollifier.