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