Mixing constructions with infinite invariant measure and spectral multiplicities

TL;DR

This paper introduces new infinite measure preserving transformations that are mixing and demonstrates their ability to realize specific spectral multiplicity sets, construct power weakly mixing transformations, and create Poissonian automorphisms with simple spectrum.

Contribution

It develops high staircase infinite measure preserving transformations and applies them to realize spectral multiplicity sets and construct various mixing automorphisms.

Findings

Realized any subset of natural numbers as spectral multiplicity set.

Constructed mixing power weakly mixing transformations.

Built mixing Poissonian automorphisms with simple spectrum.

Abstract

We introduce high staircase infinite measure preserving transformations and prove that they are mixing under a restricted growth condition. This is used to (i) realize each subset $E\subset\Bbb N\cup\{\infty\}$ as the set of essential values of the multiplicity function for the Koopman operator of a mixing ergodic infinite measure preserving transformation, (ii) construct mixing power weakly mixing infinite measure preserving transformations, (iii) construct mixing Poissonian automorphisms with a simple spectrum, etc.