Criteria of Divergence Almost Everywhere in Ergodic Theory

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–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 L^p, 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 L^p, 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.