Some factors of nonsingular Bernoulli shifts

TL;DR

This paper constructs factors of nonsingular Bernoulli shifts, showing they can be equivalent to IID systems under certain conditions and demonstrating the existence of type-III:1 Bernoulli shifts with all ergodic indices.

Contribution

It provides elementary constructions of factors of nonsingular Bernoulli shifts and answers an open question about the existence of type-III:1 Bernoulli shifts with all ergodic indices.

Findings

Nonsingular Bernoulli shifts satisfying the Doeblin condition have IID factors.

Existence of type-III:1 Bernoulli shifts with every ergodic index.

Elementary constructions of factors of nonsingular Bernoulli shifts.

Abstract

We give elementary constructions of factors of nonsingular Bernoulli shifts. In particular, we show that all nonsingular Bernoulli shifts on a finite number of symbols which satisfy the Doeblin condition have a factor that is equivalent to an independent and identically distributed system. We also prove that there are type-III:1 Bernoulli shifts of every possible ergodic index, answering a question of Danilenko and Lemanczyk (Ergodic Theory Dynam. Systems, 39(12):3292-3321, 2019).