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Ergodic theorem in CAT(0) spaces in terms of inductive means

Part of: Metric geometry Ergodic theory

Published online by Cambridge University Press:  17 March 2022

JORGE ANTEZANA Open the ORCID record for JORGE ANTEZANA [Opens in a new window] ,
EDUARDO GHIGLIONI  and
DEMETRIO STOJANOFF
JORGE ANTEZANA*
Affiliation:
Instituto Argentino de Matemática ‘Alberto P. Calderón’ (IAM-CONICET), CABA, Argentina and Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Buenos Aires, Argentina (e-mail: eghiglioni@mate.unlp.edu.ar, demetrio@mate.unlp.edu.ar)
EDUARDO GHIGLIONI
Affiliation:
Instituto Argentino de Matemática ‘Alberto P. Calderón’ (IAM-CONICET), CABA, Argentina and Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Buenos Aires, Argentina (e-mail: eghiglioni@mate.unlp.edu.ar, demetrio@mate.unlp.edu.ar)
DEMETRIO STOJANOFF
Affiliation:
Instituto Argentino de Matemática ‘Alberto P. Calderón’ (IAM-CONICET), CABA, Argentina and Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Buenos Aires, Argentina (e-mail: eghiglioni@mate.unlp.edu.ar, demetrio@mate.unlp.edu.ar)
*
e-mail: antezana@mate.unlp.edu.ar
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Abstract

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.

Keywords

ergodic theoreminductive meansbarycenterHadamard spacenon-positively curved space

MSC classification

Primary: 37A30: Ergodic theorems, spectral theory, Markov operators 51F99: None of the above, but in this section
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 5 , May 2023 , pp. 1433 - 1454
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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