FURTHER DISCUSSION ON KATO'S CHAOS IN SET-VALUED DISCRETE SYSTEMS

For a compact metric space $ Y $ and a continuous map $ g:Y\rightarrow Y $, the collective accessibility and collectively Kato chaotic of the dynamical system $ (Y, g) $ were defined. The relations between topologically weakly mixing and collective accessibility, or strong accessibility, or strongly Kato chaos were studied. Some common properties of $ g $ and $ \overline{g} $ were given. Where $ \overline{g}: \kappa(Y)\rightarrow \kappa(Y) $ is defined as $ \overline{g}(B) = g(B) $ for any $ B\in\kappa(Y) $, and $ \kappa(Y) $ is the collection of all nonempty compact subsets of $ Y $. Moreover, it is proved that $ g $ is collectively accessible (or strongly accessible) if and only if $ \overline{g} $ in $ w^{e} $-topology is collectively accessible (or strongly accessible).