Special α-Limit Points for Maps of the Interval

Hero, Michael W.

doi:10.2307/2159483

Cited by 13 publications

(27 citation statements)

References 4 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%

“…Implication (A) is proved by Theorem 1; implication (B) by Theorem 2 and the final one (C) follows from the transitivity. The counterexample for the converse implication of (A) was given by M. W. Hero [7] in Example 1. For the counterexample of the converse to (B) see Theorem 3 and for the final one (C) see Remark 2.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…Many interesting properties were proven by M. W. Hero [7] on sα-limit sets for continuous maps on the unit closed interval, see the following proposition. Here, a point x ∈ X is called almost periodic if for a given open set U containing x, one can find an integer n > 0 such that for any integer q > 0 there is an integer r, q ≤ r ≤ q + n with f r (x) ∈ U.…”

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

confidence: 96%

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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: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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

confidence: 96%

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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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“…We denote the set of all α-limit sets by α f . Although α-limit sets have not been studied quite as extensively as there ω counterparts, interest in them has been growing (see, for example, [2,17,18,28,29]).…”

Section: Introductionmentioning

confidence: 99%

Shadowing, internal chain transitivity and α-limit sets

Good

Meddaugh

Mitchell

2020

Journal of Mathematical Analysis and Applications

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“…Hero [11] studied the sets of special α-limit points for interval maps and proved the following theorem.…”

Section: Introductionmentioning

confidence: 99%