Equicontinuous mappings on finite trees

TL;DR

This paper characterizes equicontinuous maps on finite trees through eight equivalent conditions, linking topological, ultrafilter, and Ramsey-theoretic properties, thereby generalizing and complementing previous results in the field.

Contribution

It provides a comprehensive set of equivalent conditions for equicontinuity on finite trees, connecting various mathematical concepts and extending prior work.

Findings

Equicontinuity is equivalent to no arc being strictly contained in its image under some iterate.

Equicontinuity relates to the continuity of functions induced by nonprincipal ultrafilters.

Failure of equicontinuity corresponds to the discontinuity of all elements in the Ellis remainder.

Abstract

If $X$ is a finite tree and $f \colon X \longrightarrow X$ is a map, as the Main Theorem of this paper we find eight conditions, each of which is equivalent to the fact that $f$ is equicontinuous. To name just a few of the results obtained: the equicontinuity of $f$ is equivalent to the fact that there is no arc $A \subseteq X$ satisfying $A \subsetneq f^n[A]$ for some $n\in \mathbb{N}$. It is also equivalent to the fact that for some nonprincial ultrafilter $u$, the function $f^u \colon X \longrightarrow X$ is continuous (in other words, failure of equicontinuity of $f$ is equivalent to the failure of continuity of $every$ element of the Ellis remainder $g\in E(X,f)^*$). One of the tools used in the proofs is the Ramsey-theoretic result known as Hindman's theorem. Our results generalize the ones shown by Vidal-Escobar and Garc\'ia-Ferreira, and complement those of Bruckner and Ceder, Mai, and Camargo, Rinc\'on and Uzc\'ategui.