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

TL;DR

This paper constructs a simpler example of a fixed-point-free map on a tree-like continuum, with uniformly bounded valence maps, improving understanding of fixed points in such spaces.

Contribution

It introduces a new fixed-point-free map on a tree-like continuum with bounded valence, simplifying previous complex constructions.

Findings

Constructed a fixed-point-free map with bounded valence

Simplified the structure of the inverse limit space

Enhanced visualization and computational potential

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 $T_n$, $n \in \mathbb{N}$ 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.