The function ω ƒ on simple n-ods

Abstract

Given a discrete dynamical system (X, ƒ), we consider the function ωƒ-limit set from X to 2x as ωƒ(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 ωƒ 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 ωƒ when the phase space is a n-od simple T. We prove that if ωƒ 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 ωƒ is a continuous set valued function iff the family {ƒ0, ƒ1, ƒ2,} is equicontinuous. As a consequence of our results concerning the ωƒ 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 ωƒ is continuous. (2) The set of all fixed points of ƒ2 is nonempty and connected.

Keywords

• simple triod
• equicontinuity
• ω-limit set
• fixed points
• discrete dynamical system