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Topological complexity

  • F BLANCHARD (a1), B HOST (a2) and A MAASS (a3)
    • Published online: 01 June 2000

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.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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