Joint ergodicity for functions of polynomial growth

TL;DR

This paper establishes necessary and sufficient conditions for joint ergodicity in systems of measure-preserving transformations involving Hardy field functions of polynomial growth, advancing ergodic theory techniques.

Contribution

It improves existing methods for analyzing multiple ergodic averages along Hardy field functions and extends results to more general iterates combining Hardy and tempered functions.

Findings

Derived conditions for joint ergodicity with polynomial growth functions

Enhanced techniques for multiple ergodic averages

Extended analysis to linear combinations of Hardy and tempered functions

Abstract

We provide necessary and sufficient conditions for joint ergodicity results for systems of commuting measure preserving transformations for an iterated Hardy field function of polynomial growth. Our method builds on and improves recent techniques due to Frantzikinakis and Tsinas, who dealt with multiple ergodic averages along Hardy field functions; it also enhances an approach introduced by the authors and Ferr\'e Moragues to study polynomial iterates. The more general expression, in which the iterate is a linear combination of a Hardy field function of polynomial growth and a tempered function, is studied as well.