Saturated Subgroups of the Circle Group

Abstract

Consider the imbedding of ℤ into L ¹(S ¹, µ), µ a probability measure, given by n → z ⁿ. The collection {z ⁿ : n ∈ ℤ} is discrete in L ¹(X, Ɓ, µ) if and only if

$$\mathop {\lim \;\sup }\limits_{n \to \infty } \left| {\hat \mu \left( n \right)} \right| < 1.$$

This result, due to C. C. Moore and K. Schmidt [3], shows that such a measure µ is non-rigid, hence not supported on a Dirichlet set. Such measures may be viewed as being full in some sense even in the case where µ is singular. More generally, given two probability measures µ and υ on S ¹, one can map, for each n, z ⁿ in L ¹(S ¹, µ) to z ⁿ in L ¹(S ¹, µ), and seek conditions under which this map extends to a continuous homomorphism between the closures of characters in the respective spaces. We will answer this question in this chapter and discuss its relation to subgroups of the circle group such as the eigenvalue group or the group of quasi-invariance of a measure. As we saw in the previous chapters, such subgroups occur naturally in non-singular dynamics.