Equicontinuous mappings on finite trees
Volume 254 / 2021
Abstract
If $X$ is a finite tree and $f \colon X \rightarrow X$ is a map, in the Main Theorem of this paper (Theorem 1.8), we find eight conditions, each of which is equivalent to $f$ being equicontinuous. To name just a few of the results obtained: the equicontinuity of $f$ is equivalent to there being no arc $A \subseteq X$ satisfying $A \subsetneq f^n[A]$ for some $n\in \mathbb {N}$. It is also equivalent to the statement that for some nonprincipal ultrafilter $u$, the function $f^u \colon X \rightarrow X$ 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 García-Ferreira (2019), and complement those of Bruckner and Ceder (1992), Mai (2003) and Camargo, Rincón and Uzcátegui (2019).
