Orders of Oscillation Motivated by Sarnak's Conjecture--Part II

TL;DR

This paper investigates the linear disjointness between higher-order oscillating sequences and various nonlinear dynamical systems, extending previous work and introducing new concepts like multi-linearly disjoint sequences.

Contribution

It proves linear disjointness results for oscillating sequences of specific orders with polynomial skew products and systems with minimal mean attractability, and introduces multi-linearly disjoint sequences with examples.

Findings

Oscillating sequences of order m are linearly disjoint from certain polynomial skew products.

Oscillating sequences of order d are linearly disjoint from minimal mean attractable systems.

Examples of multi-linearly disjoint sequences are constructed.

Abstract

This work is a continuation of [13]. We study the linear disjointness between higher-order oscillating sequences and nonlinear dynamical systems. Specifically, we prove that any oscillating sequence of order $m=d+k-1$ and any simple polynomial skew product of degree $k$ on the $d$-Euclidean space are linearly disjoint. Additionally, we demonstrate that any oscillating sequence of order $d$ and any minimal mean attractable and minimal quasi-discrete spectrum dynamical system of order $d$ are linearly disjoint. Finally, we introduce multi-linearly disjoint sequences and construct examples of such sequences.