A CAT(0)-valued pointwise ergodic theorem
Abstract
In this note we prove the a pointwise ergodic theorem for functions taking values in a separable complete CAT(0)-space, analogous to Lindenstrauss' pointwise ergodic theorem for real-valued integrable functions on a probability space subject to a probability-preserving action of an amenable l.c.s.c. group, where in the CAT(0) setting the role of ergodic averages is played by the barycentres of the empirical distributions of a CAT(0)-valued map along an orbit of the group action. The proof rests on an approximation argument and an appeal to that result for real-valued maps.
Keywords
Cite
@article{arxiv.0905.0515,
title = {A CAT(0)-valued pointwise ergodic theorem},
author = {Tim Austin},
journal= {arXiv preprint arXiv:0905.0515},
year = {2016}
}
Comments
7 pages; [TDA, April 12st 2011]: Modified slightly following referee reports, and a small mistake corrected; [v5:] This preprint has been re-written to correct to a mistake in the proof of Lemma 2.3. The journal published that correction in a separate erratum