Type: Article

The function w f on simple n-ods

Vidal-Escobar, Ivon; García-Ferreira, Salvador;

Abstract:

Given a discrete dynamical system (X, ƒ), we consider the function ?ƒ-limit set from X to 2x as ?ƒ(x) = {y ? X : there exists a sequence of positive integers n1 < n2 < … such that limk?? ƒnk (x) = y}, for each x ? X. In the article [1], A. M. Bruckner and J. Ceder established several conditions which are equivalent to the continuity of the function ?ƒ where ƒ: [0,1] ? [0,1] is continuous surjection. It is natural to ask whether or not some results of [1] can be extended to finite graphs. In this direction, we study the function ?ƒ when the phase space is a n-od simple T. We prove that if ?ƒ is a continuous map, then Fix(ƒ2) and Fix(ƒ3) are connected sets. We will provide examples to show that the inverse implication fails when the phase space is a simple triod. However, we will prove that: Theorem A 2. If ƒ: T ? T is a continuous function where T is a simple triod then ?ƒ is a continuous set valued function iff the family {ƒ0, ƒ1, ƒ2,} is equicontinuous. As a consequence of our results concerning the ?ƒ function on the simple triod, we obtain the following characterization of the unit interval. Theorem A 1. Let G be a finite graph. Then G is an arc iff for each continuous function ƒ: G ? G the following conditions are equivalent: (1) The function ?ƒ is continuous. (2) The set of all fixed points of ƒ2 is nonempty and connected.