Special $$\alpha $$ α -Limit Points and $$\gamma $$ γ -Limit Points of a Dendrite Map

Sun, Taixiang; Tang, Yalin; Su, Guangwang; Xi, Hongjian; Qin, Bin

doi:10.1007/s12346-017-0225-4

Cited by 8 publications

(7 citation statements)

References 18 publications

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“…[18], [3]). It was shown in [31] that Rec(f ) ⊆ SA(f ) holds in the special case of maps acting on dendrites with countable set of endpoints and that there are dendrite maps with SA(f ) ⊂ Rec(f ). We show that Rec(f ) ⊆ SA(f ) holds for general dynamical systems in Corollary 2.2.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the structure of $α$-limit sets of backward trajectories for graph maps

Foryś-Krawiec,

Hantáková,

Oprocha

2021

Preprint

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In the paper we study what sets can be obtained as α-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those α-limit sets are ω-limit sets and for all but finitely many points x, we can obtain every ω-limits set as the α-limit set of a backward trajectory starting in x. For zero entropy maps, every α-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On the structure of $α$-limit sets of backward trajectories for graph maps

Foryś-Krawiec,

Hantáková,

Oprocha

2021

Preprint

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“…Before turning our attention to Question 1.2, and in keeping with the precedent set in [23], we will briefly highlight a corollary concerning γ-limit sets. These were introduced by Hero [27] who studied them for interval maps, γ-limit sets have since been further examined by Sun et al in [46] and [45] for graph maps and dendrites respectively. The γ-limit set of a point x, denoted γ(x), is defined by saying that, for any y ∈ X, y ∈ γ(x) if and only if y ∈ ω(x) and there exists a sequence y i ∞ i=1 in X and a strictly increasing sequence n i ∞ i=1 in N such that f n i (y i ) = x for each i and lim i→∞ y i = y.…”

Section: γ-Limit Sets and Ict Fmentioning

confidence: 99%

“…This presents a difficulty when attempting to give a suitable definition for α f (x). Various approaches to this have been taken [2,16,17,27,45,46]. For a discussion on this, we refer the reader to [23].…”

Section: Introductionmentioning

confidence: 99%

When is the beginning the end? On full trajectories, limit sets and internal chain transitivity

Mitchell

2022

Topology and its Applications

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“…[19], [4]). It was shown in [33] that Rec(f ) ⊆ SA(f ) holds in the special case of maps acting on dendrites with countable set of endpoints and that there are dendrite maps with SA(f ) ⊂ Rec(f ). We show that Rec(f ) ⊆ SA(f ) holds for general dynamical systems in Corollary 1.…”

mentioning

confidence: 99%

On the structure of $ \alpha $-limit sets of backward trajectories for graph maps

Foryś-Krawiec¹,

Hantáková²,

Oprocha³

2022

DCDS

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<p style='text-indent:20px;'>In the paper we study what sets can be obtained as <inline-formula><tex-math id="M2">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-limit sets are <inline-formula><tex-math id="M4">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>-limit sets and for all but finitely many points <inline-formula><tex-math id="M5">\begin{document}$ x $\end{document}</tex-math></inline-formula>, we can obtain every <inline-formula><tex-math id="M6">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>-limits set as the <inline-formula><tex-math id="M7">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-limit set of a backward trajectory starting in <inline-formula><tex-math id="M8">\begin{document}$ x $\end{document}</tex-math></inline-formula>. For zero entropy maps, every <inline-formula><tex-math id="M9">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations.</p>

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Special $$\alpha $$ α -Limit Points and $$\gamma $$ γ -Limit Points of a Dendrite Map

Cited by 8 publications

References 18 publications

On the structure of $α$-limit sets of backward trajectories for graph maps

On the structure of $α$-limit sets of backward trajectories for graph maps

When is the beginning the end? On full trajectories, limit sets and internal chain transitivity

On the structure of $ \alpha $-limit sets of backward trajectories for graph maps

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