Continuous maps with the whole space being a scrambled set
Let \((X,d)\) be a metric space and let \(f:X\rightarrow X\) be a continuous map. Essentially following \textit{T.-Y. Li} and \textit{J. A. Yorke} [Am. Math. Mon. 82, 985-992 (1975; Zbl 0351.92021)], a subset \(S\) of \(X\) containing at least two elements is called a scrambled set of \(f\) if for any \(x,y\in S\) with \(x\neq y\), \(\limsup_{n\rightarrow \infty} d(f^n(x),f^n(y))>0\) and \(\liminf_{n\rightarrow \infty} d(f^n(x),f^n(y))=0\). The map \(f\) is said to be chaotic (in the sense of Li and Yorke) if it has an uncountable scrambled set. The problem of whether continuous maps may possess large scrambled sets has been widely discussed in the literature. In particular, examples of chaotic maps having scrambled sets of full Lebesgue measure have been provided by \textit{M. Misiurewicz} [Lect. Notes Math. 1163, 125-130 (1985; Zbl 0625.58007)] and \textit{A. M. Bruckner} and \textit{T. Hu} [Trans. Am. Math. Soc. 301, 289-297 (1987; Zbl 0639.26004)] in the case \(X=[0,1]\) and by \textit{I. Mizera} [Dynamical systems and ergodic theory, Banach Cent. Publ. 23, 161-164 (1989; Zbl 0701.58046)] in the case \(X=[0,1]^n\) for any \(n\geq 2\). All maps above are just continuous; \textit{H. Kato} [Proc. Am. Math. Soc. 126, No. 7, 2151-2157 (1998; Zbl 0893.54033)] has recently shown that there are homeomorphisms with similar properties for any \(n\) (except in the obvious case \(n=1\)). The paper under review continues this line of work providing for any \(n\geq 2\) a homeomorphism from the open cube \((0,1)^n\) into itself for which the whole space is a scrambled set; analogous examples [replacing ``\((0,1)\)'' by ``\((0,\infty)\)'' or ``\((-\infty, \infty)\)'' and ``homeomorphism'' by ``\(C^\infty\) diffeomorphism'', now also including the case \(n=1\))] have just been constructed by the same author [Trans. Am. Math. Soc. 351, 353-362 (1999; Zbl 0924.58051)].
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