Equicontinuous mappings on finite trees

Abstract

If $X$ is a finite tree and $f:X⟶X$ 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 $A⊊f^n[A]$ for some $n\in \mathbb{N}$. It is also equivalent to the fact that for some nonprincial ultrafilter $u$, the function $f^u:X⟶X$ is continuous (in other words, failure of equicontinuity of $f$ is equivalent to the failure of continuity of $every$ element of the Ellis remainder $g\in E(X,f{)}^{\ast }$). 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 García-Ferreira, and complement those of Bruckner and Ceder, Mai, and Camargo, Rincón and Uzcátegui.

30 pages, 2 figures; minor typos corrected from the previous version