The function ω ƒ on simple n-ods

Given a discrete dynamical system (X, ƒ), we consider the function ω ƒ -limit set from X to 2 x as ω ƒ (x) = {y ∈ X : there exists a sequence of positive integers  n 1 < n 2 < … such that lim k → ∞ ƒ nk (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 ω 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 ω ƒ when the phase space is a n-od simple T. We prove that if ω ƒ is a continuous map, then Fix(ƒ 2 ) and Fix(ƒ 3 ) 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 ω ƒ is a continuous set valued function iff the family {ƒ 0 , ƒ 1 , ƒ 2 ,} is equicontinuous. As a consequence of our results concerning the ω 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 ω ƒ is continuous. (2) The set of all fixed points of ƒ 2 is nonempty and connected.