×

The nonexistence of expansive homeomorphisms of chainable continua. (English) Zbl 0868.54032

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)

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
PDFBibTeX XMLCite
Full Text: DOI EuDML