de Mathématiques de Luminy
Ferenczi Sébastien, Holton
Charles and Zamboni Luca Q.
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 threeinterval 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 measuretheoretic 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 interesting examples of weakly mixing three-interval exchanges. Finally we give affirmative answers to two long standing questions posed by W.A. Veech on the existence of three-interval exchanges having irrational eigenvalues and discrete spectrum.