Relativization of complexity and sensitivity

• Autores: Guohua Zhang
• Localización: Ergodic theory and dynamical systems, ISSN 0143-3857, Vol. 27, Nº 4, 2007, págs. 1349-1371
• Idioma: inglés
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• Resumen

  • First notions of relative complexity function and relative sensitivity are introduced. It turns out that for any open factor map $\pi: (X, T)\rightarrow (Y, S)$ between topological dynamical systems with minimal $(Y, S),\ \pi$ is positively equicontinuous if and only if the relative complexity function is bounded for each open cover of $X$; and that any non-trivial weakly mixing extension is relatively sensitive. Moreover, a relative version of the notable result that any $M$-system is sensitive if it is not minimal is obtained. Then notions of relative scattering and relative Mycielski's chaos are introduced. A relative disjointness theorem involving relative scattering is given. A relative version of the well-known result that any non-trivial scattering topological dynamical system is Li-Yorke chaotic is proved.