Abstract
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 ∈ L1 (X, μ), and to compare the global average ∫ f dμ of f to the pointwise averages ∣D∣−1 ∑δ∈Df(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, λΓ) 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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Abért and B. Weiss, Bernoulli actions are weakly contained in any free action, Ergodic Theory and Dynamical Systems 33 (2013), 323–333.
M. A. Akcoglu and A. del Junco, Convergence of averages of point transformations, Proceedings of the American Mathematical Society 49 (1975), 265–266.
N. Alon and J. H. Spencer, The Probabilistic Method, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 2000.
C. Anantharaman, J.-P. Anker, M. Babillot, A. Bonami, B. Demange, S. Grellier, F. Havard, P. Jaming, E. Lesigne, P. Maheux, J.-P. Otal, B. Schapira and J.-P. Schreiber, Théorèmes ergodiques pour les actions de groupes, Monographies de L’Enseignement Mathématique, Vol. 41, L’Enseignement Mathématique, Geneva, 2010.
J. Beck, An algorithmic approach to the Lovász local lemma, Random Structures & Algorithms 2 (1991), 343–365.
A. Bernshteyn, Measurable versions of the Lovász local lemma and measurable graph colorings, Advances in Mathematics 353 (2019), 153–223.
A. Bernshteyn, Building large free subshifts using the local lemma, Groups, Geometry, and Dynamics, to appear, https://doi.org/abs/1802.07123.
G. D. Birkhoff, Proof of the ergodic theorem, Proceedings of the National Acadamey of Sciences of the United States of America 17 (1931), 656–660.
L. Bowen and A. Nevo, Pointwise ergodic theorems beyond amenable groups, Ergodic Theory and Dynamical Systems 33 (2013), 777–820.
A. Bufetov and A. Klimenko, On markov operators and ergodic theorems for group actions, European Journal of Combinatorics 33 (2012), 1427–1443.
P. J. Burton, Topology and convexity in the space of actions modulo weak equivalence, Ergodic Theory and Dynamical Systems 38 (2018), 2508–2536.
P. J. Burton and A. S. Kechris, Weak containment of measure preserving group actions, Ergodic Theory and Dynamical Systems, to appear, https://doi.org/10.1017/etds.2019.26.
C. Conley, S. Jackson, A. Marks, B. Seward and R. Tucker-Drob, Hyperfiniteness and Borel combinatorics, Journal of the European Mathematical Society, to appear, https://doi.org/abs/1611.02204.
E. Csóka, L. Grabowski, A. Máthé, O. Pikhurko and K. Tyros, Borel version of the local lemma, https://doi.org/abs/1605.04877.
A. del Junco and J. Rosenblatt, Counterexamples in ergodic theory and number theory, Mathematische Annalen 245 (1979), 185–197.
P. Erdős and L. Lovász, Problems and results on 3-chromatic hypergraphs and some related questions, in Infinite and Finite Sets, Colloquia Mathematica Societatis Janos Bolyai, Vol. 10, North-Holland, Amsterdam, 1975, pp. 609–627.
E. Glasner, J.-P. Thouvenot and B. Weiss, Every countable group has the weak Rohlin property, Bulletin of the London Mathematical Society 138 (2006), 932–936.
H. Hatami, L. Lovász and B. Szegedy, Limits of locally-globally convergent graph sequences, Geometric and Functional Analysis 24 (2014), 269–296.
A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, Vol. 156, Springer, New York, 1995.
A. S. Kechris, Global Aspects of Ergodic Group Actions, Mathematical Surveys and Monographs, Vol. 160, American Mathematical Society, Providence, RI, 2010.
A. S. Kechris and B. D. Miller, Topics in Orbit Equivalence, Lecture Notes in Mathematics, Vol. 1852, Springer, Berlin, 2004.
K. Kolipaka and M. Szegedy, Moser and Tardos meet Lovász, in STOC’11—Proceedings of the 43rd ACM Symposium on Theory of Computing ACM, New York, 2011, pp. 235–244.
G. Kun, Expanders have a spanning Lipschitz subgraph with large girth, https://doi.org/abs/1303.4982.
E. Lindenstrauss, Pointwise theorems for amenable groups, Inventiones Mathematicae 146 (2001), 259–295.
M. Molloy and B. Reed, Graph Colouring and the Probabilistic Method, Algorithms and Combinatorics, Vol. 23, Springer, Berlin, 2002.
R. Moser and G. Tardos, A constructive proof of the general Lovász local lemma, Journal of the ACM 57 (2010), Article no. 11.
A. Rumyantsev and A. Shen, Probabilistic constructions of computable objects and a computable version of Lovász local lemma, Fundamenta Informaticae 132 (2014), 1–14.
J. H. Spencer, Asymptotic lower bounds for Ramsey functions, Discrete Mathematics 20 (1977), 69–76.
A. Tempelman, Ergodic Theorems for Group Actions, Mathematics and its Applications, Vol. 78, Kluwer Academic Publishers, Dordrecht, 1992.
R. D. Tucker-Drob, Weak equivalence and non-classifiability of measure preserving actions, Ergodic Theory and Dynamical Systems 35 (2015), 293–336.
J. von Neumann, Proof of the quasi-ergodic hypothesis, Proceedings of the National Academy of Sciences of the United States of America 18 (1932), 70–82.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research is supported in part by the Waldemar J., Barbara G., and Juliette Alexandra Trjitzinsky Fellowship.
Rights and permissions
About this article
Cite this article
Bernshteyn, A. Ergodic theorems for the shift action and pointwise versions of the Abért-Weiss theorem. Isr. J. Math. 235, 255–293 (2020). https://doi.org/10.1007/s11856-019-1957-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-019-1957-4