Ergodic Theory and Dynamical Systems

Topological complexity

a1 IML-CNRS, case 907, 163 avenue de Luminy, 13288 Marseille cedex 09, France (e-mail:
a2 Équipe d'analyse et de mathématiques appliquées, Université de Marne-la-Vallée, 5 Boulevard Descartes, Champs sur Marne, 77454 Marne-la-Vallée cedex, France (e-mail:
a3 Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170/3 correo 3, Santiago, Chile (e-mail:


In a topological dynamical system $(X,T)$ the complexity function of a cover ${\cal C}$ is the minimal cardinality of a sub-cover of $\bigvee_{i=0}^n T^{-i}{\cal C}$. 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 $(x,y)$, $x\ne y$, 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.

(Received April 10 1998)
(Revised January 10 1999)