Topological complexity, return times and weak disjointness

In Blanchard et al (Topological complexity. Ergod. Th. & Dynam. Sys.20 (2000), 641–662), the authors introduced the notion of scattering and a weaker notion of 2-scattering. It is an open question whether the two notions are equivalent. The question is answered affirmatively in this paper. Using the complexity function of an open cover along some sequences of natural numbers, we characterize mild mixing, strong scattering and scattering. We show that mildly mixing (respectively strongly mixing) systems are disjoint from minimal uniformly rigid (respectively minimal rigid) systems.Moreover, assuming minimality we show that a dynamical system is full-scattering (respectively mildly mixing or weakly mixing) if and only if it is strongly mixing (respectively IP*-transitive or with the lower Banach density 1.