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Difference between Devaney chaos associated with two systems. (English) Zbl 1191.37010
The paper discusses the existence of the Devaney chaos in two dynamical systems defined on compact metric spaces. One of the considered systems is a so-called hyperspace system, which is associated with the other system called the base system. The hyperspace system is defined on the space of all non-empty compact sets of the base space and endowed with the Hausdorff metric. The Devaney chaos is characterized by the topological transitivity, the sensitive dependence on the initial conditions, and the dense distribution of the periodic orbits. The latter condition is examined for the two considered systems. It is shown that this condition can be satisfied for the hyperspace system, whereas it does not hold for the underlying base system. This phenomenon is demonstrated for the mixing symbolic dynamics of the hyperspace and base systems, where the latter is taken as a shift map on the space of infinite sequences composed of three symbols. The paper thus concludes that the hyperspace system being Devaney chaos need not imply the base system being Devaney chaos.
37B10Symbolic dynamics
54H20Topological dynamics