Equicontinuous mappings on finite trees

Abstract

If $X$ is a finite tree and $f \colon X \longrightarrow X$ is a continuous function, as the Main Theorem of this paper (Theorem 8), we find eight conditions, each of which is equivalent to the fact that $f$ is equicontinuous. Our results either generalize ones shown by Vidal-Escobar and Garcia-Ferreira, or complement those of Bruckner and Ceder, Mai and Camargo, Rincon and Uzcategui. Some of the methods, however, have not been used previously in this context (for example, in one of our proofs we apply the Ramsey-theoretic result known as Hindman's theorem). 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 $E^*(X,f)$).