On the functions ωf and Ωf

Javier Camargo; Johan Cancino; Carlos Islas

doi:10.1515/ms-2024-0057

Article

On the functions ω_f and Ω_f

Javier Camargo , Johan Cancino and Carlos Islas

Published/Copyright: June 24, 2024

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From the journal Mathematica Slovaca Volume 74 Issue 3

Abstract

Given a compactum X, a map f : X → X and x ∈ X, it is defined ω(x, f) = {y ∈ X : there exists an increasing sequence (n_i)_i∈ℕ ⊆ ℕ such that limi→∞ f^n_i(x) = y}; and Ω(x, f) = {y ∈ X : there exist an increasing sequence (n_i)_i∈ℕ ⊆ ℕ and a sequence (x_i)_i∈ℕ in X such that limi→∞ x_i = x and limi→∞ f^n_i(x_i) = y}. It is well known that both ω(x, f) and Ω(x, f) are nonempty compact subsets of X. 2^X denotes the collection of all the nonempty compact subsets of X endowed with the Hausdorff metric. In this paper, we consider ω_f : X → 2^X and Ω_f : X → 2^X defined for each x ∈ X by ω_f(x) = ω(x, f) and Ω_f(x) = Ω(x, f), respectively. We study the continuity of these functions, when f is defined on some kind of continua.

MSC 2010: Primary 54H20; 37B45

Keywords: Continuum; continuum of the type λ; dendrites; ω-limit function; equicontinuous function; periodic points

The first author thanks La Vicerrectoría de Investigación y Extensión de la Universidad Industrial de Santander y su Programa de Movilidad for financial support. Also, the first author thanks La Universidad Autónoma de la Ciudad de México for the partial support during this research. The third author thanks to the project CCH-ACA-022-004-2 of the Universidad Autónoma de la Ciudad de México for the financial support.

Acknowledgement

The authors thank the referee for her/his valuable comments to improve the paper.

Communicated by L’ubica Holá

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Received: 2023-05-03

Accepted: 2023-10-20

Published Online: 2024-06-24

Published in Print: 2024-06-25

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https://doi.org/10.1515/ms-2024-0057

Keywords for this article

Continuum; continuum of the type λ; dendrites; ω-limit function; equicontinuous function; periodic points