Criteria of Divergence Almost Everywhere in Ergodic Theory

TL;DR

This paper surveys classical tools like the Stein continuity principle, entropy criteria, and Kakutani-Rochlin lemma used to analyze divergence almost everywhere in ergodic theory, providing new proofs and connections to Gaussian and stable processes.

Contribution

It offers a comprehensive exposition of divergence criteria in ergodic theory, including new applications and detailed proofs for entropy and Kakutani-Rochlin based divergence conditions.

Findings

Established an $L^1$-version of the continuity principle and its application to Fourier series divergence.

Provided detailed proofs of entropy criteria in $L^p$, $2 ext{ extminus} o ext{ extminus} ext{ extgreater} ext{ extgreater} extgreater}

Linked maximal operators with Gaussian processes and studied $p$-stable processes for divergence in $L^p$.

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.