Fixed points in indecomposable $k$-junctioned tree-like continua

Charles L. Hagopian
2010 Proceedings of the American Mathematical Society  
Let M be an indecomposable k-junctioned tree-like continuum. Let f be a map of M that sends each composant of M into itself. Using an argument of O. H. Hamilton, we prove that f has a fixed point. D. P. Bellamy [1] in 1979 defined a tree-like continuum that does not have the fixed-point property (also see [4], [10], [11], and [13]). In 1998, the author [6] proved that every tree-like continuum has the fixed-point property for deformations. This was accomplished by showing that every map of a
more » ... t every map of a tree-like continuum M that sends each arc-component of M into itself has a fixed point. The collection of arc-components is a partition of a continuum. Hence a map that sends each arc-component into itself is sometimes called an arc-component-preserving map. Bellamy's continuum is indecomposable and its arc-components are composants. The collection of composants of an indecomposable continuum is a partition that is refined by the collection of arc-components. The difference between these two partitions is sometimes extreme. Each arc-component consists of one point in P. Minc's example [11] of a hereditarily indecomposable tree-like continuum without the fixed-point property. Must every composant-preserving map of an indecomposable tree-like continuum M have a fixed point? We prove the answer is yes if M is k-junctioned.
doi:10.1090/s0002-9939-09-10165-x fatcat:oydnwu6qmffstfnmte7hwpy3ti