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Using an example of D. P. Bellamy, we define a 3-composant treelike continuum admitting a fixed-point-free homeomorphism that sends each composant onto itself. This continuum is used to define an indecomposable tree-like continuum that admits a composant-preserving fixed-point-free homeomorphism. A map extension theorem is proved and applied in this construction. Suppose X is a plane continuum and f is a map of X that sends each arccomponent of X into itself. In 1988, Hagopian [3] proved that f has a fixed point if X is tree-like or indecomposable. Solenoids show this is not true for nonplanar indecomposable continua. However, in 1998 Hagopian [4] proved every tree-like continuum has the fixed-point property for arc-component-preserving maps. Recently, Hagopian [5] proved that every composant-preserving map of an indecomposable k-junctioned tree-like continuum has a fixed point. He used tree-chain covers to get his results. Question 1 of [5] asks if this theorem can be generalized to every indecomposable tree-like continuum. We use an example of Bellamy [1] to answer this question in the negative. In fact, we define an indecomposable tree-like continuum that admits a composant-preserving fixed-point-free homeomorphism. It is not known if there exists a tree-like plane continuum or an indecomposable plane continuum that admits a composant-preserving fixed-point-free map. A continuum is a nonempty compact connected metric space. A continuum is indecomposable if it is not the union of two proper subcontinua. Let x be a point of a nondegenerate continuum X. The x-composant of X is the union of all proper subcontinua of X that contain x. If X is indecomposable, then X is the union of uncountably many dense disjoint composants. Given > 0, a mapping f : X → Y is an -mapping if diam(f−1(y)) < for each y ∈ Y . A continuum X is tree-like if for each > 0, there exist a tree Y and an -mapping of X onto Y . A continuum X is tree-like if and only if for each > 0 there is an -tree-chain covering X. A tree-like continuum X is k-junctioned if k is the least integer such that for every positive number there is an -tree-chain covering X with k junction links. We refer to a locally compact noncompact metric space P as a parameter space and denote by P ∪ {∞} the one-point compactification of P . A map α : P → P of a parameter space to itself is called infinity preserving provided that whenever Received by the editors December 8, 2010. 2010 Mathematics Subject Classification. Primary 54F15, 54H25.

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