On the relationship between zero entropy and quasi-discrete spectrum for affine transformations

Abstract

Introduction. If T is a measure-preserving transformation of a Lebesgue space (M, (3, m) let 7r(T) be the maximum partition such that T.(T) has zero entropy. In ?2 we prove a result, which was stated without proof in [12], concerning the behaviour of the partitions 7(Tr.) associated with an increasing sequence of invariant measurable partitions {f ,n}. If T is an ergodic affine transformation of a compact connected metric abelian group G let (T) be the maximum partition such that T,,(T) has quasi-discrete spectrum. In ?3 we prove, using the result of ?2 and a method introduced by Rohlin in [8], that q(T) -7r(T). This result was first obtained by Parry [6] who also proved that the maximum partition y(T) such that T7(T) has the distal property is also 7r(T). My thanks are due to Dr. W. Parry for his guidance during this work.