Rigidity times for weakly mixing dynamical system which are not rigidity times for any irrational rotation

TL;DR

This paper constructs a specific sequence and measure to demonstrate the existence of a weakly mixing dynamical system with rigidity times that are not shared by any system with discrete spectrum, highlighting a novel distinction in rigidity properties.

Contribution

It introduces a new sequence and measure to show that certain rigidity times in weakly mixing systems are not rigidity times for any systems with discrete spectrum.

Findings

Constructed a sequence dense in the circle for irrational rotations.

Established a measure with specific rigidity properties.

Proved the existence of a weakly mixing system with unique rigidity times.

Abstract

We construct an increasing sequence of natural numbers $(m_n)_{n=1}^{+\infty}$ with the property that $(m_n \th [1])_{n\geq 1}$ is dense in $\T$ for any $\th \in \R\setminus \Q$, and a continuous measure on the circle $\mu$ such that $\lim_{n\to +\infty}\int_{\T}\|m_n\theta\|d\mu(\theta)=0$. Moreover, for every fixed $k\in \N$, the set $\{n\in \N:\,k\nmid m_n \}$ is infinite. This is a sufficient condition for the existence of a rigid, weakly mixing dynamical system whose rigidity time is not a rigidity time for any system with a discrete part in its spectrum.