Several transitive properties and Devaney's chaos (Q270138)

Let \((X,f)\) be a dynamical system. The following are the main results of this paper. Theorem 1. If \(f\) is a homeomorphism, then \(f\) is indecomposable iff \(f\) is \(Z\)-transitive. Theorem 2. If \(X\) has no isolated points, then \(f\) is \(Z\)-transitive iff \(f\) is transitive. Theorem 3. The following statements are equivalent: {\parindent=8mm \begin{itemize}\item[(i)] \(f\) is Devaney's chaos \item[(ii)] \(f\) is weakly transitive and \(\overline{P(f)}=X\) \item[(iii)] \(f\) is \(Z\)-transitive and \(\overline{P(f)}=X\), \end{itemize}} Theorem 4. Suppose that \(f\) has either the \(\underline{d}\)-shadowing property or the \(\overline{d}\)-shadowing property. Then, for any \(k\in N\), \(f^k\) is weakly transitive. Further technical aspects of these results are also discussed.