Ergodic Theorems for the Shift Action and Pointwise Versions of The Ab\'ert--Weiss Theorem

Abstract

Let $\Gamma$ be a countably infinite group. A common theme in ergodic theory is to start with a probability measure-preserving (p.m.p.) action $\Gamma \curvearrowright (X, \mu)$ and a map $f \in L^1(X, \mu)$, and to compare the global average $\int f \,\mathrm{d}\mu$ of $f$ to the pointwise averages $|D|^{-1} \sum_{\delta \in D} f(\delta \cdot x)$, where $x \in X$ and $D$ is a nonempty finite subset of $\Gamma$. The basic hope is that, when $D$ runs over a suitably chosen infinite sequence, these pointwise averages should converge to the global value for $\mu$-almost all $x$. In this paper we prove several results that refine the above basic paradigm by uniformly controlling the averages over specific sets $D$ rather than considering their limit as $|D| \to \infty$. Our results include ergodic theorems for the Bernoulli shift action $\Gamma \curvearrowright ([0;1]^\Gamma, \lambda^\Gamma)$ and strengthenings of the theorem of Ab\'ert and Weiss that the shift is weakly contained in every free p.m.p. action of $\Gamma$. In particular, we establish a purely Borel version of the Ab\'ert--Weiss theorem for finitely generated groups of subexponential growth. The central role in our arguments is played by the recently introduced measurable versions of the Lov\'asz Local Lemma, due to the current author and to Cs\'oka, Grabowski, M\'ath\'e, Pikhurko, and Tyros.