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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 T ¯ on the space 𝒦(𝕏) 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 T ¯ is zero or infinity. Moreover, the topological entropy of T ¯| 𝒞(𝕏) is zero, where 𝒞(𝕏) 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:
37B40Topological entropy
37B99Topological dynamics
37E05Maps of the interval (piecewise continuous, continuous, smooth)
37E10Maps of the circle
37B10Symbolic dynamics