On the Zero-Type property and mixing of Bernoulli shifts

TL;DR

This paper investigates the properties of Bernoulli shifts, proving a dichotomy between zero-type and the existence of an equivalent invariant stationary product probability, with examples of complex mixing behaviors.

Contribution

It establishes a fundamental classification for non-singular Bernoulli shifts and provides examples of shifts with power weakly mixing and zero-type properties.

Findings

Non-singular Bernoulli shifts are either zero-type or have an equivalent invariant stationary product probability.

Constructed examples of Bernoulli shifts and Markovian flows that are power weakly mixing and zero type.

Abstract

We prove that every non-singular Bernoulli shift is either zero-type or there is an equivalent invariant stationary product probability. We also give examples of a type Bernoulli shift and a Markovian flow which are power weakly mixing and zero type.