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Criteria of Divergence Almost Everywhere in Ergodic Theory

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In this expository paper, we survey nowadays classical tools or criteria used in problems of convergence everywhere to build counterexamples: the Stein continuity principle, Bourgain’s entropy criteria, and Kakutani–Rokhlin lemma, the most classical device for these questions in ergodic theory. First, we state a L1-version of the continuity principle and give an example of its usefulness by applying it to a famous problem on divergence almost everywhere of Fourier series. Next we particularly focus on entropy criteria in Lp, 2 ≤ p ≤ ∞, and provide detailed proofs. We also study the link between the associated maximal operators and the canonical Gaussian process on L2. We further study the corresponding criterion in Lp, 1 < p < 2, using properties of pstable processes. Finally, we consider Kakutani–Rokhlin’s lemma, one of the most frequently used tools in ergodic theory, by stating and proving a criterion for a.e. divergence of weighted ergodic averages. Bibliography: 38 titles.

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Authors and Affiliations

  1. IRMA, Université Louis-Pasteur et C.N.R.S., Strasbourg, France

    M. J. G. Weber

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  1. M. J. G. Weber
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Correspondence to M. J. G. Weber.

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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 441, 2015, pp. 73–118.

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Weber, M.J.G. Criteria of Divergence Almost Everywhere in Ergodic Theory. J Math Sci 219, 651–682 (2016). https://doi.org/10.1007/s10958-016-3137-y

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