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Topological entropy for set valued maps. (English) Zbl 1193.37019
Summary: Any continuous map $T$ on a compact metric space $\Bbb X$ induces in a natural way a continuous map $\overline T$ on the space $\cal K(\Bbb X)$ of all non-empty compact subsets of $\Bbb X$. Let $T$ be a homeomorphism on the interval or on the circle. It is proved that the topological entropy of the induced set valued map $\overline T$ is zero or infinity. Moreover, the topological entropy of $\overline T|_{\cal C(\Bbb X)}$ is zero, where $\cal C(\Bbb X)$ denotes the space of all non-empty compact and connected subsets of $\Bbb X$. For general continuous maps on compact metric spaces these results are not valid.

MSC:
37B40Topological entropy
37B99Topological dynamics
37E05Maps of the interval (piecewise continuous, continuous, smooth)
37E10Maps of the circle
37B10Symbolic dynamics
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References:
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