TY - JOUR
AU - Vidal-Escobar, Ivon
AU - Garcia-Ferreira, Salvador
PY - 2019/10/01
Y2 - 2024/09/18
TI - The function ω ƒ on simple n-ods
JF - Applied General Topology
JA - Appl. Gen. Topol.
VL - 20
IS - 2
SE -
DO - 10.4995/agt.2019.11065
UR - https://polipapers.upv.es/index.php/AGT/article/view/11065
SP - 325-347
AB - <p>Given a discrete dynamical system (X, ƒ), we consider the function ω<sub>ƒ</sub>-limit set from X to 2<sup>x </sup>as</p><p>ω<sub>ƒ</sub>(x) = {y âˆˆ X : there exists a sequence of positive integers <br /> n<sub>1</sub> < n<sub>2</sub> < ... such that lim<sub>k</sub><sub>â†’</sub><sub>âˆž</sub> ƒ<sup>nk</sup> (x) = y},</p><p>for each x âˆˆ X. In the article [1], A. M. Bruckner and J. Ceder established several conditions which are equivalent to the continuity of the function ω<sub>ƒ</sub> where ƒ: [0,1] â†’ [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 ω<sub>ƒ</sub> when the phase space is a n-od simple T. We prove that if ω<sub>ƒ</sub> is a continuous map, then Fix(ƒ<sup>2</sup>) and Fix(ƒ<sup>3</sup>) 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:</p><p>Theorem A 2. If ƒ: T â†’ T is a continuous function where T is a simple triod then ω<sub>ƒ</sub> is a continuous set valued function iff the family {ƒ<sup>0</sup>, ƒ<sup>1</sup>, ƒ<sup>2</sup>,} is equicontinuous.</p><p>As a consequence of our results concerning the ω<sub>ƒ</sub> function on the simple triod, we obtain the following characterization of the unit interval.</p><p>Theorem A 1. Let G be a finite graph. Then G is an arc iff for each continuous function ƒ: G â†’ G the following conditions are equivalent:<br /> (1) The function ω<sub>ƒ</sub> is continuous.<br /> (2) The set of all fixed points of ƒ<sup>2 </sup>is nonempty and connected.</p>
ER -