E. H. El Abdalaoui
Published 2014-06-10, updated 2016-03-03Version 2
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.
Comments: The purpose of this new version is to correct the previous version of this work by proving that the well-known open problem of the pointwise convergence of the Furstenberg ergodic averages has a positive answer if we restrict our self to the convergence along a subsequence. Consequently, we provide an example of non-singular transformation for which the maximal ergodic inequality does not hold
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