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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.

##### MSC:
 37C05 Dynamical systems involving smooth mappings and diffeomorphisms
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##### References:
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