Almost sure convergence of the multiple ergodic average for certain weakly mixing systems

TL;DR

This paper proves that for weakly mixing PID systems, the multiple ergodic averages converge almost surely, resolving a long-standing open problem in ergodic theory.

Contribution

It establishes almost sure convergence of multiple ergodic averages for weakly mixing PID systems, extending known results in ergodic theory.

Findings

Almost sure convergence of multiple ergodic averages in weakly mixing PID systems

Positive answer to a long-standing open problem in ergodic theory

Advances understanding of mixing implications for multiple averages

Abstract

The family of pairwise independently determined (PID) systems, i.e. those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin, of whether mixing implies mixing of all orders, has a positive answer. We show that in the class of weakly mixing PID one finds a positive answer for another long-standing open problem, whether the multiple ergodic averages \begin{equation*} \frac 1 N\sum_{n=0}^{N-1}f_1(T^nx)\cdots f_d(T^{dn}x), \quad N\to \infty, \end{equation*} almost surely converge.