Tree-like continua with invariant composants under fixed-point-free homeomorphisms (Q2845881)

A (metric) continuum \(X\) has the fixed point property (fpp) if for each continuous map \(f\) from \(X\) into itself, there exists a point \(p\) in \(X\) such that \(f(p)=p\). In 1979, \textit{D. P. Bellamy} [Houston J. Math. 6, 1--14 (1980; Zbl 0447.54039)] published the classical example of a tree-like continuum without the fpp. Since then several authors have been trying to find tree-like continua without the fpp and with additional properties. In [Fundam. Math. 155, No. 2, 161--176 (1998; Zbl 0918.54034)] \textit{C. Hagopian} proved that every tree-like continuum has the fpp for arc-component-preserving maps and in [Proc. Am. Math. Soc. 138, No. 4, 1511--1515 (2010; Zbl 1192.54017)] he asked if every composant-preserving map of an indecomposable tree-like continuum has a fixed point. In the paper under review, the authors show that the fpp continues giving unexpected examples. Using Bellamy's example they construct: (a) a 3-composant tree-like continuum admitting a fixed-point-free homeomorphism that sends each composant into itself; and (b) an indecomposable tree-like continuum that admits a composant-preserving fixed-point-free homeomorphism.