A self-map of a tree-like continuum with no invariant indecomposable subcontinuum
A very important step is made towards enlightening the behaviour of fixed-point-free maps on tree-like continua. All known examples of such maps -- in particular the first one due to \textit{D. P. Bellamy} [Houston J. Math. 6, 1-13 (1980; Zbl 0447.54039)], where the continuum has been indecomposable, including the author's examples [e.g. Trans. Am. Math. Soc. 351, No. 3, 1109-1121 (1999; Zbl 0917.54040)], transform some indecomposable subcontinua into itself as in the hypothetical fixed-point-free maps of continua non-separating the plane. The author shows that in general this is not true. He shows even more. For every natural number \(j\) he constructed a tree-like continuum \(Y\) and a self-map on it such that none of the first \(j\) iterates of this map transforms any indecomposable subcontinuum of \(Y\) -- including single points -- into itself, the further iterates however have fixed points.
scientific article; zbMATH DE number 1367565