Topological Entropy and Mixing Invariant Extremal Distributional Chaos
We show that there exists a mixing dynamical system with an invariant, extremal and transitive distributionally scrambled set. Meanwhile, we prove that such a complex dynamical system has zero topological entropy. Finally, we extend the method of constructing the “strong” distributionally scrambled set and show that the new dynamical system has a positive topological entropy.
Dense invariant open distributionally scrambled sets and closed distributionally scrambled sets
Abstract It is known that in a compact dynamical system, the whole space can be a Li–Yorke scrambled set, but this does not hold for distributional chaos. In this paper we prove that the complement of a distributionally scrambled set must be an infinite set. Then we give an example of an uncountable dense invariant open extremal distributionally scrambled set which is the complement of a countable infinite set. This presents one kind of the “largest” (from the topological point of view) distributionally scrambled set in a compact dynamical system. Moreover, we construct an uncountable closed distributionally scrambled set.
Set-valued discrete chaos
Abstract Given a continuous map f : X → X on a metric space (X, d), we characterize topological transitivity for the (set-valued) map f ¯ : K ( X ) → K ( X ) induced by f on the space K ( X ) of compact subsets of X, endowed with the Hausdorff distance. More precisely, f ¯ is transitive if and only if f is weakly mixing. Some consequences are also derived for the dynamics on fractals and for (continuous and) linear maps on infinite-dimensional spaces.