A tree-like continuum admitting fixed point free maps with arbitrarily small trajectories
To answer a problem posed by \textit{R. H. Bing} [Am. Math. Mon. 76, 119-132 (1969; Zbl 0174.259)], \textit{D. P. Bellamy} [Houston J. Math. 6, 1-13 (1980; Zbl 0447.54039)] constructed a tree-like continuum without the fixed point property. The author modifies Bellamy's construction to obtain a tree-like continuum \(X\) with the following property: For every \(\varepsilon>0\) and every positive integer \(j\) there is a map \(f:X\to X\) such that \(f^ i(x)\neq x\) for each \(x\in X\) and \(1\leq i\leq j\), and the diameter of each trajectory \(\bigcup_{m=-\infty}^ \infty f^ m(x)\) is smaller than \(\varepsilon\).
scientific article; zbMATH DE number 96237
scientific article