Kato, Hisao The nonexistence of expansive homeomorphisms of chainable continua. (English) Zbl 0868.54032 Fundam. Math. 149, No. 2, 119-126 (1996). A homeomorphism \(h : X \to X\) of a compactum \(X\) is said to be expansive if there exists \(c > 0\) such that for any distinct \(x, y \in X\) there exists \(n \in Z\) with dist\((h^n(x), h^n(y)) > c.\) The author shows that if \(X\) is a continuum and \(h : X \to X\) is semiconjugate to a continuous surjection of the pseudo-arc \(P\) then \(h\) is not expansive. By a result of this reviewer [Proc. Am. Math. Soc. 91, 147-154 (1984; Zbl 0553.54018)] this implies that no nondegenerate chainable continuum admits an expansive homeomorphism. It is not known whether there is a nondegenerate tree-like or hereditarily indecomposable continuum which admits an expansive homeomorphism. R. Williams has conjectured that no nondegenerate nonseparating plane continuum admits an expansive homeomorphism. The author [Can. J. Math. 45, No. 3, 576-598 (1993; Zbl 0797.54047)] has shown that such chainable continua as the Knaster “buckethandle” and the pseudo-arc do admit continuum-wise expansive homeomorphisms. Reviewer: W.Lewis (Lubbock) Cited in 16 Documents MSC: 54H20 Topological dynamics (MSC2010) 54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites 54B20 Hyperspaces in general topology 54E40 Special maps on metric spaces Keywords:expansive homeomorphism; chainable continuum; pseudo-arc; hereditarily indecomposable continuum; hyperspace Citations:Zbl 0553.54018; Zbl 0797.54047 PDFBibTeX XMLCite \textit{H. Kato}, Fundam. Math. 149, No. 2, 119--126 (1996; Zbl 0868.54032) Full Text: DOI EuDML