### 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

#### Subject classification

54H20; 54E40; 37B45.

PDF

#### References

A. M. Bruckner and J. Ceder, Chaos in terms of the map x → ω(x,f)\$, Pacific J. Math. 156 (1992), 63-96. https://doi.org/10.2140/pjm.1992.156.63

R. Gu, Equicontinuity of maps on figure-eight space, Southeast Asian Bull. Math. 25 (2001), 413-419. https://doi.org/10.1007/s100120100004

W. Hurewicz and H. Wallman, Dimension theory, Princeton University Press, Princeton (1941). https://doi.org/10.1515/9781400875665

A. Illanes and S. B. Nadler, Jr., Hyperspaces: Fundamental and Recent Advances, A Series of Monographs and Textbooks Pure and Applied Mathematics 216, Marcel Decker Inc. New York (1998).

S. Kolyada and L. Snoha, Some aspects of topological transitivity a survey, Grazer Math. Ber. 334 (1997), 3-35.

L. Lum, A Characterization of Local Connectivity in Dendroids, Studies in Topology (Proc. Conf., Univ. North Carolina, Charlotte NC 1974); Academic Press (1975), 331-338. https://doi.org/10.1016/B978-0-12-663450-1.50033-5

J. Mai, The structure of equicontinuous maps, Trans. Amer. Math. Soc. 355 (2003), 4125-4136. https://doi.org/10.1090/S0002-9947-03-03339-7

S. B. Nadler, Jr., Continuum Theory: An Introduction, A Series of Monographs and Textbooks Pure and Applied Mathematics 158, Marcel Decker Inc. New York (1992).

T. X. Sun, G. W. Su, H. J. Xi and X. Kong, Equicontinuity of maps on a dendrite with finite branch points, Acta Math. Sin. (Engl. Ser.) 33 (2017), 1125-1130. https://doi.org/10.1007/s10114-017-6289-x

Abstract Views

1254

#### Cited-By (articles included in Crossref)

This journal is a Crossref Cited-by Linking member. This list shows the references that citing the article automatically, if there are. For more information about the system please visit Crossref site

1. The ω-limit function on dendrites
Javier Camargo, Johan Cancino
Topology and its Applications  vol: 282  first page: 107320  year: 2020
doi: 10.1016/j.topol.2020.107320

 Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.Universitat Politècnica de Valènciae-ISSN: 1989-4147   https://doi.org/10.4995/agt