Blanchard, F.; Host, B.; Maass, A. Topological complexity. (English) Zbl 0962.37003 Ergodic Theory Dyn. Syst. 20, No. 3, 641-662 (2000). A dynamical system \((X,T)\) given by a compact metric space \(X\) and a continuous map \(T\) from \(X\) onto \(X\) is considered. Given a finite cover \({\mathcal C}\) of \(X\) the minimal cardinality of a subcover of \({\mathcal C}\) is denoted by \(r({\mathcal C})\). The authors define the complexity of the finite cover \({\mathcal C}\) as the nondecreasing function \(c({\mathcal C},n)= r(\bigvee_{i=0}^n T^{-i}{\mathcal C})\). It is shown that the system is equicontinuous if and only if for any finite open cover \({\mathcal C}\) of \(X\), the complexity \(c({\mathcal C},n)\) is bounded. A system is called scattering if any finite cover by non-dense open sets has unbounded complexity. If this is true for all standard covers, i.e. the covers consisting of two non-dense open sets, the system is 2-scattering. It is proved that all topologically weakly mixing systems are scattering. If for every standard cover \({\mathcal C}\), \(c({\mathcal C},n)> n+1\) for some \(n\), then the system is topologically weakly mixing. Further it is shown that all 2-scattering systems are totally transitive. For a minimal dynamical system 2-scattering, scattering and weak mixing are equivalent. A system is scattering if and only if its Cartesian product with any minimal system is transitive. As a consequence, scattering systems are disjoint from all minimal distal systems. A pair of points \((x,y)\), \(x\neq y\) is said to be a complexity pair if every standard cover \({\mathcal C}= (A,B)\) that separates \(x\) and \(y\) (i.e., \(x\in \text{Int}(A^c)\) and \(y\in \text{Int} (B^c))\) has unbounded complexity. Properties of complexity pairs imply that all 2-scattering systems are disjoint from minimal equicontinuous systems (minimal equicontinuous systems are conjugate to minimal isometries). It is shown that there are strong links between complexity pairs and regionally proximal pairs. If \(T\) is a homeomorphism then the complexity relation is contained in the regionally proximal relation. If the homeomorphism is minimal, then these two relations coincide up to the diagonal. Remarks about determinism and chaos are added. Reviewer: Ľubomír Snoha (Banská Bystrica) Cited in 5 ReviewsCited in 61 Documents MSC: 37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) 54H20 Topological dynamics (MSC2010) Keywords:topological complexity; scattering; transitivity; minimality; equicontinuity; distality; disjointness × Cite Format Result Cite Review PDF Full Text: DOI