Mean equicontinuity, almost automorphy and regularity

The aim of this article is to obtain a better understanding and classification strictly ergodic topological dynamical systems with (measurable) discrete spectrum. To that end, we first determine when an isomorphic maximal equicontinuous factor map minimal system has trivial (one point) fibers. In other words, characterize mean are almost automorphic. Furthermore, investigate another natural subclass systems, so-called diam-mean show if only the regular (the points fibers have full Haar measure). Combined previous results in field, provides characterization for every step hierarchy models We also construct example transitive positive entropy, give partial answer question Furstenberg related multiple recurrence.