Structure of three-interval exchange transformations III: Ergodic and spectral properties

In this paper, we present a detailed study of the spectral/ergodic properties of three-interval exchange transformations. Our approach is mostly combinatorial and relies on the diophantine results in Part I and the combinatorial description in Part II. We define a recursive method of generating three sequences of nested Rokhlin stacks which describe the system from a measure-theoretic point of view and which in turn gives an explicit characterization of the eigenvalues. We obtain necessary and sufficient conditions for weak mixing which, in addition to unifying all previously known examples, allow us to exhibit new examples of weakly mixing three-interval exchanges. Finally, we give affirmative answers to two questions posed by W. A. Veech on the existence of three-interval exchanges having irrational eigenvalues and discrete spectrum.[PUBLICATION ABSTRACT]