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- W2886820122 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 Abert 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 Abert--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 Lovasz Local Lemma, due to the current author and to Csoka, Grabowski, Mathe, Pikhurko, and Tyros." @default.
- W2886820122 created "2018-08-22" @default.
- W2886820122 creator A5019172041 @default.
- W2886820122 date "2018-08-01" @default.
- W2886820122 modified "2023-09-27" @default.
- W2886820122 title "Ergodic Theorems for the Shift Action and Pointwise Versions of The Ab'ert--Weiss Theorem" @default.
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