Pointwise ergodic theorem for locally countable quasi-pmp graphs

Anush Tserunyan

doi:10.3934/jmd.2022019

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2022, Volume 18: 609-655. Doi: 10.3934/jmd.2022019

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Pointwise ergodic theorem for locally countable quasi-pmp graphs

Anush Tserunyan

Mathematics and Statistics Department, McGill University, Montréal, Quebec H3A 0B9, Canada

Received: September 29, 2019

Revised: May 21, 2022

The author's research was partially supported by NSF Grant DMS-1501036.

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Abstract

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, equivalently, Schreier graphs of quasi-pmp actions of countable groups. For ergodic graphs, the theorem gives an increasing sequence of Borel subgraphs with finite connected components over which the averages of any $ L^1 $ function converges to its expectation. This implies that every (not necessarily pmp) locally countable ergodic Borel graph on a standard probability space contains an ergodic hyperfinite subgraph. A consequence of this is that every ergodic treeable equivalence relation has an ergodic hyperfinite free factor.

The pmp case of the main theorem was first proven by R. Tucker-Drob using a deep result from probability theory. Our proof is different: it is self-contained and applies more generally to quasi-pmp graphs. Among other things, it involves introducing a graph invariant concerning asymptotic averages of functions and a method of tiling a large part of the space with finite sets with prescribed properties. The non-pmp setting additionally exploits a new quasi-order called visibility to analyze the interplay between the Radon–Nikodym cocycle and the graph structure, providing a sufficient condition for hyperfiniteness.

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Mathematics Subject Classification: Primary: 37A30; Secondary: 03E15, 05C63, 37A20, 37A25.

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