The function ω ƒ on simple n-ods

Ivon Vidal-Escobar, Salvador Garcia-Ferreira

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

Subject classification

54H20; 54E40; 37B45.

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References

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