## A note on transitivity in set-valued discrete systems.(English)Zbl 1098.37008

Summary: Let $$(X,d)$$ be a metric space and $$f:X \to X$$ is a continuous function. If we consider the space $$(\mathcal K(X),H)$$ of all non-empty compact subsets of $$X$$ endowed with the Hausdorff metric induced by $$d$$ and $$\bar f : \mathcal K(X) \to \mathcal K(X)$$, $$\bar f(A) = \{f(a)/a \in A\}$$, then the aim of this work is to show that $$\bar f$$ transitive implies $$f$$ transitive. Also, we give an example showing that $$f$$ transitive does not implies $$\bar f$$ transitive.

### MSC:

 37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) 54H20 Topological dynamics (MSC2010)
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### References:

 [1] Banks, J.; Brooks, J.; Cairns, G.; Stacey, P., On the devaney’s definition of chaos, Am. math. monthly, 99, 332-334, (1992) · Zbl 0758.58019 [2] Devaney, R.L., An introduction to chaotic dynamical systems, (1989), Addison-Wesley Redwood City · Zbl 0695.58002 [3] Klein, E.; Thompson, A., Theory of correspondences, (1984), Wiley-Interscience New York [4] Román-Flores, H.; Barros, L.C.; Bassanezi, R.C., A note on the zadeh’s extensions, Fuzzy sets syst., 17, 327-331, (2001) · Zbl 0968.54007 [5] Román-Flores, H., Discrete fuzzy dynamical systems: a first approach, Rev. mat. ing., 9, 35-42, (2001) [6] Vellekoop, M.; Berglund, R., On intervals, transitivity=chaos, Am. math. monthly, 101, 4, 353-355, (1994) · Zbl 0886.58033
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