Ergodic theorems for the shift action and pointwise versions of the Abért-Weiss theorem

• Published in: Israel journal of mathematics 2020, Vol.235 (1), p.255-293
• Main Author: Bernshteyn, Anton
• Format: Article
• Language: English
• Subjects: Algebra; Analysis; Applications of Mathematics; Ergodic processes; Group Theory and Generalizations; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Theorems; Theoretical
• ISSN: 0021-2172; 1565-8511
• DOI: 10.1007/s11856-019-1957-4

Let Γ be a countably infinite group. A common theme in ergodic theory is to start with a probability measure-preserving (p.m.p.) action Γ ↷ ( X, μ ) and a map f ∈ L 1 ( X, μ ), and to compare the global average ∫ f d μ of f to the pointwise averages ∣ D ∣ −1 ∑ δ ∈ D f ( δ · x ), where x ∈ X and D is a nonempty finite subset of Γ. The basic hope is that, when D runs over a suitably chosen infinite sequence, these pointwise averages should converge to the global value for μ -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 ∣ → ∞. Our results include ergodic theorems for the Bernoulli shift action Γ ↷ ([0; 1] Γ , λ Γ ) and strengthenings of the theorem of Abért and Weiss that the shift is weakly contained in every free p.m.p. action of Γ. In particular, we establish a purely Borel version of the Abért–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 Lovász Local Lemma, due to the current author and to Csóka, Grabowski, Máthé, Pikhurko, and Tyros.