A note on distributional chaos with respect to a sequence
The paper contains new results on distributional chaos - a generalization of the chaos in the sense of Li and Yorke. Let \((X,d)\) be a compact metric space and \(f:X\mapsto X\) a continuous onto map. Let \(W^{(k)}\subset X^k\) be a closed \(f^{\times k}\)-invariant subset such that \(\Delta_X^{(k)}\subset W^{(k)}\) and \(f^{\times k}\mid_{W^{(k)}}\) is transitive (here \(f^{\times k}\) is the \(k\)-times Cartesian product of \(f\) and \(\Delta_X^{(k)}:=\{(\chi,\dots,\chi)\in X^k:\chi\in X\}\) is diagonal in \(X^k\)). If there is a perfect set \(S\subset X\) such that \(W^{(k)}\cap S^k\) is residual in \(S^k\) for every \(k=2,3,\dots\) then there is an increasing sequence \(\{p_i\}_{i\in N}\) and a Micielski set \(M\subset S\) dense in \(S\) being distributionally scrambled with respect to \(\{p_i\}_{i\in N}\). This result will ensure that \(f\) is distributionally chaotic with respect to \(\{p_i\}_{i\in N}\), in the sense introduced in this paper.
scientific article; zbMATH DE number 5613005
scientific article