Mixing automorphisms not having the Kolmogorov spectral property

TL;DR

This paper investigates the spectral properties of a specific class of mixing automorphisms, showing they lack the Kolmogorov spectral property and analyzing their tensor products.

Contribution

It demonstrates that certain staircase rank-one automorphisms with parameters growing as j^d do not have the group spectral property and have simple spectrum in their tensor products.

Findings

Spectral group property is absent for these automorphisms.

Tensor product T⊗T has simple spectrum.

Automorphisms exhibit specific spectral behaviors based on parameter growth.

Abstract

Let $T$ be a staircase rank-one construction with parameters $r_j \sim j^d$, $0<d<0.2$, then its spectrum does not have the group property, and the symmetric tensor product $T\odot T$ has simple spectrum.