K-property for Maharam extensions of nonsingular Bernoulli and Markov shifts

TL;DR

This paper classifies conservative nonsingular Bernoulli shifts into types and shows that their Maharam extensions are K-automorphisms in the type III_1 case, extending previous results and constructing new examples.

Contribution

It extends classification results for nonsingular Bernoulli shifts and their Maharam extensions, including new constructions for type III_1 shifts and generalizations to Markov shifts.

Findings

Conservative nonsingular Bernoulli shifts are either type II_1 or III_1.

Maharam extensions of type III_1 shifts are K-automorphisms.

New examples of nonequilibrial shifts of type III_1 are constructed.

Abstract

It is shown that each conservative nonsingular Bernoulli shift is either of type $II_1$ or $III_1$. Moreover, in the latter case the corresponding Maharam extension of the shift is a $K$-automorphism. This extends earlier results obtained by Z.~Kosloff for the equilibrial shifts. Nonequilibrial shifts of type $III_1$ are constructed. We further generalize (partly) the main results to nonsingular Markov shifts.