Homeomorphisms with the whole compacta being scrambled sets (Q2709591)

This paper studies some properties of chaotic dynamical systems, and more precisely, completely scrambled homeomorphisms on infinite compact metric spaces. A homeomorphism on a metric space \((X,d)\) is said completely scrambled if for each \(x\neq y\), \(\limsup_{n\to\infty} d(f^n(x),f^n(y)) > 0\), and \(\liminf_{n\to\infty}d(f^n(x), f^n(y)) >0\). In this case \(f\) is proximal and chaotic in the sense of Li and Yorke [\textit{T. Li, J. Yorke}, Am. Math. Mon. 82, 985-992 (1975; Zbl 0351.92021)]. The aim of the paper under review is to construct many infinite compact metric spaces that admit completely scrambled homeomorphisms, disproving a conjecture by J. Mai on their non-existence [\textit{J. Mai}, Chin. Sci. Bull. 42, 1494-1497 (1997; Zbl 0930.37010)].NEWLINENEWLINENEWLINEThe paper first gives a characterization of the countable compact metric spaces X that admit completely scrambled homeomorphisms in terms of their derived degree \(d(X)\): a necessary and sufficient condition is that \(d(X)\) is a limit ordinal and \(X^{d(X)}\) is a set consisting of one point. Further examples of sets admitting completely scrambled homeomorphisms are construct including the Cantor set and connected compact metric spaces of arbitrary topological dimension. As a by-product, the existence of a continuous map \(g\) of [0,1] is proved such that there is an invariant scrambled set \(S\) for \(g\) contained in the set of \(\omega\)-limit points of \(g\), which is the Cantor set, and on which the restriction \(g|X\) of \(g\) is a homeomorphism.NEWLINENEWLINENEWLINEThe examples of completely scrambled homeomorphisms which are constructed have only one recurrent point, and they have zero topological entropy. The paper ends by conjecturing that there exist completely scrambled homeomorphisms with mixing properties.