Gaussian automorphisms whose ergodic self-joinings are Gaussian
We study ergodic properties of the class of Gaussian automorphisms whose ergodic self-joinings remain Gaussian. For such automorphisms we describe the structure of their factors and of their centralizer. We show that Gaussian automorphisms with simple spectrum belong to this class. We prove a new sufficient condition for non-disjointness of automorphisms giving rise to a better understanding of Furstenberg's problem relating disjointness to the lack of common factors. This and an elaborate study of isomorphisms between classical factors of Gaussian automorphisms allow us to give a complete solution of the disjointness problem between a Gaussian automorphism whose ergodic self-joinings remain Gaussian and an arbitrary Gaussian automorphism.