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.

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

References in corpus (2)

• Pointwise convergence of multiple ergodic averages and strictly ergodic models
• Pointwise recurrence for commuting measure preserving transformations

Cited by in corpus (4)

• Pointwise convergence of multiple ergodic averages and strictly ergodic models
• Pointwise recurrence for commuting measure preserving transformations
• Almost sure convergence of the multiple ergodic average for certain weakly mixing systems
• On the polynomials homogeneous ergodic bilinear averages with Liouville and Möbius weights