Joint ergodicity of Hardy field sequences

TL;DR

This paper proves the mean convergence of multiple ergodic averages involving Hardy field functions of polynomial growth, extending previous results to a broad class of functions including logarithmico-exponential functions.

Contribution

It establishes convergence and characteristic factors for ergodic averages with iterates from Hardy field functions, confirming a conjecture of Frantzikinakis.

Findings

Convergence of ergodic averages for Hardy field functions of polynomial growth.

Identification of characteristic factors under certain conditions.

Results apply to weak-mixing systems and generalize previous work.

Abstract

We study mean convergence of multiple ergodic averages, where the iterates arise from smooth functions of polynomial growth that belong to a Hardy field. Our results include all logarithmico-exponential functions of polynomial growth, such as the functions $t^{3/2}, t\log t$ and $e^{\sqrt{\log t}}$. We show that if all non-trivial linear combinations of the functions $a_1,...,a_k$ stay logarithmically away from rational polynomials, then the $L^2$-limit of the ergodic averages $\frac{1}{N} \sum_{n=1}^{N}f_1(T^{\lfloor{a_1(n)}\rfloor}x)\cdots f_k(T^{\lfloor{a_k(n)}\rfloor}x)$ exists and is equal to the product of the integrals of the functions $f_1,...,f_k$ in ergodic systems, which establishes a conjecture of Frantzikinakis. Under some more general conditions on the functions $a_1,...,a_k$, we also find characteristic factors for convergence of the above averages and deduce a convergence result for weak-mixing systems.