Self-embeddings of trees

TL;DR

This paper proves a fix point theorem for monoids of self-embeddings of trees, leading to a characterization of trees based on fixed points of their self-embeddings, with implications for their structural properties.

Contribution

It introduces a new fix point theorem for monoids of self-embeddings of trees and generalizes a previous result by Laflamme, Pouzet, and Sauer.

Findings

Trees either contain a subdivided binary tree or have fixed points under all self-embeddings.

The fix point theorem provides a new tool for analyzing tree automorphisms.

The results connect structural properties of trees with their self-embedding monoids.

Abstract

We prove a fix point theorem for monoids of self-embeddings of trees. As a corollary, we obtain a result by Laflamme, Pouzet and Sauer that a tree either contains a subdivided binary tree as a subtree or has a vertex, and edge, an end or two ends fixed by all its self-embeddings.