The function ω ƒ on simple n-ods

Abstract

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 
n₁ < n₂ < ... 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(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 ω_ƒ is a continuous set valued function iff the family {ƒ⁰, ƒ¹, ƒ²,} 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 ƒ² is nonempty and connected.