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

Rodrigo Hernández-Gutiérrez; L. C. Hoehn

paper

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

arXiv:1608.08094

Abstract

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 $T_{n}$ , $n \in N$ 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 $lim (T_{n}, f_{n})$ . 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.