# Spectral properties and approximations of joinings for infinite rank-one actions

## TL;DR

This paper investigates the spectral properties of infinite rank-one transformations, revealing complex behaviors of their joinings and spectra, including the existence of transformations with mixed spectral types depending on parameters.

## Contribution

It introduces a class of infinite rank-one transformations with polynomial weak closure, exhibiting minimal self-joinings and novel spectral phenomena.

## Key findings

- Existence of transformations with simple singular spectrum for certain powers
- Transformations with Lebesgue spectrum for other powers
- Similar spectral effects in Gaussian and Poisson suspensions

## Abstract

An ergodic self-joining of an infinite rank-one transformation is a part of the weak limit of off-diagonal measures. A class of uncountaible cardinality of nonisomorphic transformations with polynomial weak closure is presented. Such actions have minimal self-joinings and some unusual spectral properties. For any set $M$ of positive integers there exists an infinite transformation $T$ such that the products $T\otimes T^m$ have simple singular spectrum as $1<m\in M$, and Lebesgue spectrum as $1<m\notin M$. There are similar effects for the corresponding Gaussian and Poisson suspensions.