Equicontinuous mappings on finite trees

If X is a finite tree and f : X → 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 ⊆ X satisfying A ⊆ fn[A] for some n ∈ N. It is also equivalent to the statement that for some nonprincipal ultrafilter u, the function fu : 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 2 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 Garciá-Ferreira (2019), and complement those of Bruckner and Ceder (1992), Mai (2003) and Camargo, Rincón and Uzcategui (2019).