Equicontinuous mappings on finite trees

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.