Transitivity, mixing and chaos for a class of set-valued mappings. (English) Zbl 1193.37023

Summary: Consider the continuous map \(f: X\to X\) and the continuous map \(\overline(f)\) of \(K(X)\) into itself induced by \(f\), where \(X\) is a metric space and \(K(X)\) the space of all non-empty compact subsets of \(X\) endowed with the Hausdorff metric. According to the questions whether the chaoticity of \(f\) implies the chaoticity of \(\overline(f)\) posed by Román-Flores and when the chaoticity of \(f\) implies the chaoticity of \(\overline(f)\) posed by Fedeli, we investigate the relations between \(f\) and \(\overline(f)\) in the related dynamical properties such as transitivity, weakly mixing and mixing, etc. And by using the obtained results, we give the satisfied answers to Román-Flores’s question and Fedeli’s question.


37C05 Dynamical systems involving smooth mappings and diffeomorphisms
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