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Topological entropy for set valued maps. (English) Zbl 1193.37019
Summary: Any continuous map $T$ on a compact metric space $𝕏$ induces in a natural way a continuous map $\overline{T}$ on the space $𝒦\left(𝕏\right)$ of all non-empty compact subsets of $𝕏$. 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 $\overline{T}{|}_{𝒞\left(𝕏\right)}$ is zero, where $𝒞\left(𝕏\right)$ denotes the space of all non-empty compact and connected subsets of $𝕏$. For general continuous maps on compact metric spaces these results are not valid.
##### MSC:
 37B40 Topological entropy 37B99 Topological dynamics 37E05 Maps of the interval (piecewise continuous, continuous, smooth) 37E10 Maps of the circle 37B10 Symbolic dynamics