A property of ergodic flows

TL;DR

This paper introduces Property B for ergodic flows, showing it prevents certain flows from being of product type or approximately transitive, and constructs a specific non-AT flow using this property.

Contribution

It defines Property B for ergodic flows and demonstrates its implications for flow classification and construction of non-AT flows.

Findings

Flows with Property B are not of product type.

Properly ergodic flows with Property B are not approximately transitive.

Constructed a non-AT flow based on a dyadic odometer.

Abstract

In this paper we introduce a property of ergodic flows, called Property B. We prove that any ergodic hyperfinite equiva- lence relation of type III_o whose associated flow satisfies this property is not of product type. A consequence of this result is that any properly ergodic ow with Property B is not approximately transitive. We use Property B to construct a non-AT flow which - up to conjugacy - is a flow built under a function with the dyadic odometer as base automorphism.