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 Zap. Nauchn. Sem. POMI, 2015, Volume 441, Pages 73–118 (Mi znsl6228)

Criteria of divergence almost everywhere in ergodic theory

M. J. G. Weber

IRMA, Université Louis-Pasteur et C.N.R.S., 7 rue René Descartes, 67084 Strasbourg Cedex, France

Abstract: 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–Rochlin lemma, most classical device for these questions in ergodic theory. First, we state a $L^1$-version of the continuity principle and give an example of its usefulness by applying it to some famous problem on divergence almost everywhere of Fourier series. Next we particularly focus on entropy criteria in $L^p$, $2\le p\le\infty$, and provide detailed proofs. We also study the link between the associated maximal operators and the canonical Gaussian process on $L^2$. We further study the corresponding criterion in $L^p$, $1<p<2$, using properties of $p$-stable processes. Finally we consider Kakutani–Rochlin's lemma, one of the most frequently used tool in ergodic theory, by stating and proving a criterion for a.e. divergence of weighted ergodic averages.

Key words and phrases: Bourgain's entropy criteria, Stein's continuity principle, Gaussian process, stable process, metric entropy, GB set, GC set, Kakutani–Rochlin lemma.

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English version:
Journal of Mathematical Sciences (New York), 2016, 219:5, 651–682

Bibliographic databases:

UDC: 519.2
Received: 12.11.2015
Language:

Citation: M. J. G. Weber, “Criteria of divergence almost everywhere in ergodic theory”, Probability and statistics. Part 22, Zap. Nauchn. Sem. POMI, 441, POMI, St. Petersburg, 2015, 73–118; J. Math. Sci. (N. Y.), 219:5 (2016), 651–682

Citation in format AMSBIB
\Bibitem{Web15} \by M.~J.~G.~Weber \paper Criteria of divergence almost everywhere in ergodic theory \inbook Probability and statistics. Part~22 \serial Zap. Nauchn. Sem. POMI \yr 2015 \vol 441 \pages 73--118 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6228} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3504500} \transl \jour J. Math. Sci. (N. Y.) \yr 2016 \vol 219 \issue 5 \pages 651--682 \crossref{https://doi.org/10.1007/s10958-016-3137-y} 

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