Topological complexity

In a topological dynamical system (X, T) the complexity function of a cover C is the minimal cardinality of a sub-cover of V-i(n)=0 T-iC. It is Shown that equicontinuous transformations are exactly those such that any open cover has bounded complexity. Call scattering a system such that any finite cover by non-dense open sets has unbounded complexity, and call 2-scattering a system such that any such 2-set cover has unbounded complexity: then all weakly mixing systems are scattering and all 2-scattering systems are totally transitive. Conversely, any system that is not 2-scattering has covers with complexity at most n + 1. Scattering systems are characterized topologically as those such that their cartesian product with any minimal system is transitive; they are consequently disjoint from all minimal distal systems. Finally, defining (x, y), x not equal y, to be a complexity pair if any cover by two non-trivial closed sets separating x from y has unbounded complexity, we prove that 2-scattering systems are disjoint from minimal isometries; that in the invertible case the complexity relation is contained in the regionally proximal relation and, when further assuming minimality, coincides with it up to the diagonal.