A fixed point theorem for inverse limits of fans

Abstract

In [1] David Bellamy gave an example of a tree-like continuum without the fixed point property. Another example of a tree-like continuum which admits a fixed point free mapping was given in [5] by L. G. Oversteegen and J. T. Rogers, Jr. We prove in this paper that a certain inverse limit of fans has the fixed point property. Certain possible methods for simplifying Bellamy's and Oversteegen and Roger's examples are eliminated by our theorem. A continuum is a nondegenerate compact connected metric space. A continuous function will be referred to as a map or mapping. A continuum X has the fixed point property provided that whenever / is a mapping of X into X, there is a point x in X such that/(x) = x. For each i > 1, let B¡ be a O-dimensional compact subset of £' X (-1). Let z be the point of E2 with first coordinate 0 and second coordinate 1. For each a in B, U (z), let La = {ta\0 < t < 1). We also let L = Lz. Finally, we let F, = U{La| a e B¡) and T¡ = % U L. We shall refer to T¡ as a fan with isolated edge L. We note that the origin of E2, denoted 0, is the branchpoint of T¡. Theorem. Suppose that X = lim{T¡, b/}, where, for each í > 1, b\ + ' is a surjection, b] + '(0) = 0, and ifa<EBj+x, there is a c e B¡ such that b\+\La) = Lc. Then X has the fixed point property.

Characterizations of $\mathcal P$-like continua that do not have the fixed point property

“…The plane fixed point problem has been listed as the first problem among the classical problems in continuum theory, see Cook, Ingram and Lelek [19], and Prajs and Charatonik [64]. No question has attracted more interest from continua theorists than this problem (see Rogers [65, p. 305]), and many papers connected the topic have appeared since 1930, including [1,2,3,5,6,7,9,12,13,14,15,16,18,21,22,23,24,25,26,27,29,30,31,32,33,36,37,38,39,41,42,46,47,48,50,51,54,55,56,57,58,59,62,63,66,67]. In these papers there were many atte...…”