A fixed-point-free map of a tree-like continuum induced by bounded valence maps on trees

• Published in: arXiv.org 2016-08
• Main Authors: Hernández-Gutiérrez, Rodrigo, Hoehn, L C
• Format: Article
• Language: English
• Subjects: Trees
• ISSN: 2331-8422

Towards attaining a better working understanding of fixed points of maps of tree-like continua, Oversteegen and Rogers constructed a tree-like continuum with a fixed-point-free self-map, described explicitly in terms of inverse limits. Specifically, they developed a sequence of trees \(T_n\), \(n \in \mathbb{N}\) and maps \(f_n\) and \(g_n\) from \(T_{n+1}\) to \(T_n\) for each \(n\), such that the \(g_n\) maps induce a fixed-point-free self-map of the inverse limit space \(\varprojlim (T_n,f_n)\). The complexity of the trees and the valences of the maps in their example all grow exponentially with \(n\), making it difficult to visualize and compute with their space and map. We construct another such example, in which the maps \(f_n\) and \(g_n\) have uniformly bounded valence, and the trees \(T_n\) have a simpler structure.