On the Spectral Multiplicity of a Class of Finite Rank Transformations

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 $M - 1$.