Möbius disjointness for models of an ergodic system and beyond
Résumé
We give a necessary and sufficient condition (called the strong MOMO property) for a uniquely ergodic model of an ergodic measure-preserving system to have all uniquely ergodic models of the system Möbius disjoint. It follows that all uniquely ergodic models of: ergodic unipotent diffeomorphisms on nil-manifolds, discrete spectrum automorphisms, systems given by some substitutions of constant length (including the classical Thue-Morse and Rudin-Shapiro substitutions), systems determined by Kakutani sequences are Möbius disjoint. We also discuss the absence of the strong MOMO property in positive entropy systems.