On the pointwise convergence of multiple ergodic averages
Abstract
It is shown that there exist a subsequence for which the multiple ergodic averages of commuting invertible measure preserving transformations of a Lebesgue probability space converge almost everywhere provided that the maps are weakly mixing with an ergodic extra condition. The proof provides a example of non-singular dynamical system for which the maximal ergodic inequality does not hold. We further get that the non-singular strategy to solve the pointwise convergence of the Furstenberg ergodic averages fails.
Cite
@article{arxiv.1406.2608,
title = {On the pointwise convergence of multiple ergodic averages},
author = {E. H. El Abdalaoui},
journal= {arXiv preprint arXiv:1406.2608},
year = {2017}
}
Comments
The purpose of this new version is to make on the surface the two main results : There is a non-singular transformation for which the maximal ergodic inequality does not hold. Furthermore, the pointwise convergence of the Furstenberg ergodic averages has a positive answer if we restrict our self to the convergence along a subsequence. Some misprints are corrected