Criteria of divergence almost everywhere in ergodic theory

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.