A class of ergodic transformations having simple spectrum

Abstract

A class of ergodic, measure-preserving, invertible point transformations is defined, called class S. Any measurepreserving point transformation induces a unitary operator on the Hilbert space of 22-functions. A theorem is proved here which implies that the operator induced by any transformation in class S has simple spectrum. [It is then a known result that the transformations in class S have zero entropy.] Let (X, 9, ,u) be a measure space, isomorphic to the unit interval with Lebesgue measure. A measurable, measure-preserving, invertible point transformation of X is called an automorphism of (X, i, u). A class of automorphisms, called class S for brevity, is defined below (Definition (4)). The purpose of this paper is to prove the following theorem: (1) THEOREM. Let r be an automorphism in class S. Then there exist arbitrarily small sets whose characteristic functions each generate ?2(dM) under the action of the unitary operator UT, where Ur is defined by UfQ(rx) =f (x). In particular U7 has simple spectrum. (2) DEFINITION. Let H be a Hilbert space, a bounded normal operator on H. Let vE H. Let H(v) consist of the closure of the set of all elements of the form P(T, T*)v, where P(T, T*) denotes a polynomial in and T*. To say that a vector vEH generates H under the action of T means that H= H(v). (3) DEFINITION. Let t = {IAt 1 < i ? m } be a finite, ordered collection of mutually disjoint measurable sets. Then t is called a partition. The union of the members of t need not be the whole space. Let ik be a sequence of partitions with the property that for every measurable set E, there exists a sequence of sets Ek such that each Ek is a union of members of (k, and pA(E A Ek)-*O as k-oo. Then it will be said that ik--e. Here e denotes the partition of the whole space into one-point sets. (4) DEFINITION. Let r be an automorphism of (X, iY, IA), {={A i1Ii<m} a partition. If rAi=Ai+1 for 1<i_m-1, t will Received by the editors March 30, 1970. AMS 1969 subject classifications. Primary 2870; Secondary 4730.