On the spectral multiplicity of a class of finite rank transformations

Abstract

ABSTRACT. The rank M transformations, which Chacon called the simple approximations with multiplicity M, were shown by Chacon to have maximal spectral multiplicity at most M, although no example was given where this bound is attained for M > 1. In this paper, for each natural number M > 1, we show how to construct a simple approximation with multiplicity M which is ergodic and has maximal spectral multiplicity equal to M -1. Introduction.The purpose of this paper is to give a method of constructing ergodic automorphisms t: [0,1) -> [0,1) with finite rank and maximal spectral multiplicity greater than one. In particular for each natural number M > 1 we show how to' construct ergodic automorphisms which admit simple approximations with multiplicity M (in the sense of Chacon [2]) and have maximal spectral multiplicity