An L¹ ergodic theorem with values in a nonpositively curved space via a canonical barycenter map

2011/04/27 by Andrés Navas, Navas, Andrés
Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Probability (math.PR) #math.DS #math.GR #math.PR

paper · pdf · doi:10.48550/arxiv.1104.5188

Final version: to appear in Erg. Theory and Dyn. Systems

openalex publication_date 2011/04/27 · arxiv created 2011/12/20 · arxiv updated 2011/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We extend a recent result of Tim Austin (see arXiv:0905.0515) to the L¹ setting, thus providing a general version of the Birkhoff ergodic theorem for functions taking values in nonpositively curved spaces. In this setting, the notion of a Birkhoff sum is replaced by that of a barycenter along the orbit. The construction of an appropriate barycenter map is the core of this note. In particular, we solve a problem raised by K.-T. Sturm showing that local compactness for the underlying space is superfluous for the construction (this extends a result of A. Es-Sahib and H. Heinich). As a byproduct of our construction, we prove a fixed point theorem for actions by isometries on a Buseman space.

Citations

• A CAT(0)-valued pointwise ergodic theorem

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