Everywhere chaotic homeomorphisms on manifolds and \(k\)-dimensional Menger manifolds
A map \(f:X\to X\) of a metric space \((X,d)\) is said to be everywhere chaotic if \(f\) has sensitive dependence on initial conditions and, for any nonempty open sets \(U\) and \(V\) of \(X\) and any \(\varepsilon>0\), there exist points \(x\in U\) and \(y\in V\) and a positive integer \(n\geq 1\) such that \(\text{dist} (f^n(x), f^n(y))< \varepsilon\). The author gives constructive proofs to show that every compact, connected \(n\)-manifold \(M^n\) \((n\geq 2)\) and every compact, connected Menger manifold \({\mathcal M}^n\) \((n\geq 2)\) admits an everywhere chaotic homeomorphism. The construction of such homeomorphisms relies essentially on one-dimensional maps and an idea of \textit{M. Barge} and \textit{J. Martin} [Proc. Am. Math. Soc. 110, 523-525 (1990; Zbl 0714.58036)].
sensitive dependence on initial conditions
scientific article; zbMATH DE number 937152
scientific article