On negative limit sets for one-dimensional dynamics

Balibrea, Francisco; Dvornikova, G.; Lampart, Marek; Oprocha, Piotr

doi:10.1016/j.na.2011.12.030

Cited by 15 publications

(37 citation statements)

References 10 publications

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“…This notion was refined firstly by M. W. Hero [7] (see Definition 2) in such a way that only some branches of the backward orbit are considered and recently by F. Balibrea et al [1] (see Definition 4) in such a way that exactly one branch of the backward orbit is considered. The aim of this paper is to state the forcing relationships between these three different notions of the concept of the α-limit set by proving valid implications and presenting counterexamples for invalid cases.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…A complete negative trajectory of a point x ∈ X is an infinite sequence {x n } ∞ n=0 such that x 0 = x and f (x n+1 ) = x n for any n ≥ 0. The following definition was recently conceived by F. Balibrea et al [1]. Definition 4.Let (X, f ) be a dynamical system and {x n } ∞ n=0 be a complete negative trajectory of a point x ∈ X for f .…”

Section: Definition 1let (X F ) Be a Dynamical System And X ∈ Xmentioning

confidence: 99%

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A note on the Definition of a–limit Set

Balibrea¹,

Guirao²,

Lampart³

2013

Appl. Math. Inf. Sci.

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Many phenomena coming from the biology, economy, engineering are modeled using discrete dynamical systems. The concept of backward orbit is an essential concept for understanding the dynamics of the system. In the literature various definitions of the concept of the alpha-limit point (respectively set) have been historically used. The aim of this paper is to analyze the forcing relationships between them via the proof of the valid relationships and the construction of counterexamples for the converse situation in order to clarify the scenario for the computation of these objects. Moreover, we present a discrete dynamical system (X, f ) with the following paradoxical behavior: for every point x ∈ X, its alpha-limit set is equal to the whole space X; there is a complete negative trajectory of x whose alpha-limit set is equal to a fixed point; there is a complete negative trajectory of x whose alpha-limit set is equal to X.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: Definition 1let (X F ) Be a Dynamical System And X ∈ Xmentioning

confidence: 99%

A note on the Definition of a–limit Set

Balibrea¹,

Guirao²,

Lampart³

2013

Appl. Math. Inf. Sci.

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“…As stated previously, in this paper we will not define α-limit sets of individual points, instead we focus on the accumulation points of individual backward trajectories. Note that this is the approach taken in [2] and [29].…”

Section: Various Notions Of Negative Limit Setsmentioning

confidence: 99%

“…Various approaches to this difficulty have been taken; these will be discussed in more detail in Section 3. We follow the approach taken in [2] and [29], by refraining from defining such sets for individual points, but rather defining them for backward trajectories. Given a point x ∈ X an infinite sequence x i i≤0 is called a backward trajectory of x if f (x i ) = x i+1 for all i ≤ −1 and x 0 = x.…”

Section: Introductionmentioning

confidence: 99%

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Shadowing, internal chain transitivity and α-limit sets

Good

Meddaugh

Mitchell

2020

Journal of Mathematical Analysis and Applications

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Nonwandering Sets and Special $$\alpha $$-limit Sets of Monotone Maps on Regular Curves

Daghar¹,

Marzougui²

2023

Springer Proceedings in Mathematics &Amp; Statistics

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On negative limit sets for one-dimensional dynamics

Cited by 15 publications

References 10 publications

A note on the Definition of a–limit Set

A note on the Definition of a–limit Set

Shadowing, internal chain transitivity and α-limit sets

Nonwandering Sets and Special $$\alpha $$-limit Sets of Monotone Maps on Regular Curves

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