Furstenberg systems of Hardy field sequences and applications

TL;DR

This paper investigates the statistical properties of sequences generated by smooth functions with polynomial growth, showing they relate to unipotent transformations on tori and have applications in multiple ergodic theorems and recurrence for zero entropy systems.

Contribution

It characterizes Furstenberg systems of Hardy field sequences, linking them to unipotent toral transformations and establishing their disjointness from ergodic systems, with applications to multiple recurrence.

Findings

Furstenberg systems arise from unipotent transformations on tori.

Sequences of the form $(n^{3/2})$, $(n ext{log}n)$ have Furstenberg systems with specific structure.

Results imply multiple recurrence for zero entropy, non-commuting measure preserving systems.

Abstract

We study measure preserving systems, called Furstenberg systems, that model the statistical behavior of sequences defined by smooth functions with at most polynomial growth. Typical examples are the sequences $(n^\frac{3}{2})$, $(n\log{n})$, and $([n^\frac{3}{2}]\alpha)$, $\alpha\in \mathbb{R}\setminus\mathbb{Q}$, where the entries are taken $\mod{1}$. We show that their Furstenberg systems arise from unipotent transformations on finite dimensional tori with some invariant measure that is absolutely continuous with respect to the Haar measure and deduce that they are disjoint from every ergodic system. We also study similar problems for sequences of the form $(g(S^{[n^{\frac{3}{2}}]} y))$, where $S$ is a measure preserving transformation on the probability space $(Y,\nu)$, $g\in L^\infty(\nu)$, and $y$ is a typical point in $Y$. We prove that the corresponding Furstenberg systems are strongly stationary and deduce from this a multiple ergodic theorem and a multiple recurrence result for measure preserving transformations of zero entropy that do not satisfy any commutativity conditions.