Narrow Results

Let (∑, ρ) be a one-sided symbolic space (with two symbols) and σ be the shift on ∑. Denote the set of almost periodic points by A(·) and the set of weakly almost periodic points by W(·). In this paper, we prove that there exists an uncountable set J such that σ|_J is distributively chaotic in a sequence, and J⊂W(σ)-A(σ).

Let X be a separable metric space containing at least two points, and f:X→X be continuous. In this paper, we consider dynamical systems on X, and prove that topologically weakly mixing implies distributional chaos in a sequence.

Let $(X,d)$ be a compact metric space and $F=\{f_1,f_2,\dots ,f_n\}$ be an $n$-tuple of continuous selfmaps on $X$. This paper investigates Hausdorff metric Li–Yorke chaos, distributional chaos and distributional chaos in a sequence from a set-valued view. On the basis of this research, we draw the main conclusions as follows: (i) If $F$ has a distributionally chaotic pair, especially $F$ is distributionally chaotic, the strongly nonwandering set $S\mathrm{\Omega }(F)$ contains at least two points. (ii) We give a sufficient condition for $F$ to be distributionally chaotic in a sequence and chaotic in the strong sense of Li–Yorke. Finally, an example to verify (ii) is given.

In this paper, we investigate the relations between distributional chaos in a sequence and distributional chaos ($\omega$-chaos, R–T chaos, DC3, respectively). Firstly, we prove a sufficient condition that the distributional chaos is equivalent to the distributional chaos in a sequence. Besides, we prove that distributional chaos in a sequence and $\omega$-chaos (R–T chaos, DC3, respectively) do not imply each other. Finally, we give a new definition of chaos, named DC2${}^{\prime }$, which is similar to DC2, and show that DC2${}^{\prime }$ is an invariant of topological conjugacy and an iteration invariant (that is, for any integer $N>0$, $f$ is DC2${}^{\prime }$ if and only if $f^N$ is DC2${}^{\prime }$).