Pointwise multiple averages for sublinear functions

TL;DR

This paper investigates the pointwise convergence of multiple ergodic averages with sublinear iterates in measure-preserving systems without assuming commutativity, extending known results to a broader class of functions.

Contribution

It establishes convergence results for multiple ergodic averages with sublinear growth functions, including non-commuting transformations, and provides explicit limit formulas.

Findings

Convergence of single averages implies convergence of multiple averages.

Certain sublinear functions hinder convergence in general systems but work in uniquely ergodic systems.

Explicit formulas for limit functions are derived in various cases.

Abstract

For any measure preserving system $(X,\mathcal{B},\mu,T_1,\ldots,T_d),$ where we assume no commutativity on the transformations $T_i,$ $1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order $0$ and of Fej\'er functions, i.e., tempered functions of order $0.$ We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of sublinear functions which are in general bad for convergence on arbitrary systems, but they are good for uniquely ergodic systems. The case where the fastest function is linear is addressed as well, and we provide, in all the cases, an explicit formula of the limit function.