Special α-limit points and unilateral γ-limit points for graph maps

Sun, Taixiang; Xi, Hongjian; Liang, Hailan

doi:10.1007/s11425-011-4254-1

Cited by 15 publications

(18 citation statements)

References 15 publications

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“…For instance, sα(x) does not need to be closed and its isolated points are always periodic, which is in some contrast to the properties of ω(x). If we denote by SA(f ) (respectively, ω(f )) the union of α-limit sets of all backward branches (respectively, all ω-limit sets) of a map f and by Rec(f ) the set of all recurrent points of f , then Rec(f ) ⊆ SA(f ) ⊆ Rec(f ) ⊆ ω(f ), for every map f on the topological graph (see [32], cf. [18], [3]).…”

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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“…By Lemma 4.4, P Tx (x) is either a cycle of graphs or a solenoidal set Q(x). In the latter case, Q(x) ⊃ α({y i } i≤0 ) and, by results from [34], ω(f…”

Section: Preliminariesmentioning

confidence: 91%

“…It has been shown that sα-limit sets are always analytic, but not necessarily Borel [22]. If we denote by SA(f ) (respectively, ω(f )) the union of α-limit sets of all backward branches (respectively, all ω-limit sets) of a map f and by Rec(f ) the set of all recurrent points of f , then Rec(f ) ⊆ SA(f ) ⊆ Rec(f ) ⊆ ω(f ), for every map f on the topological graph (see [34], cf. [19], [4]).…”

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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“…In [28] the author defines this set as the special α-limit set of x and examines them for interval maps. These sets are investigated in [46] and [45] for graph maps and dendrites. Call this Approach 2 (A2).…”

Section: Various Notions Of Negative Limit Setsmentioning

confidence: 99%

“…First introduced by Hero [28] 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 ni (y i ) = x for each i and lim i→∞ y i = y.…”

Section: A Remark On γ-Limit Setsmentioning

confidence: 99%

Shadowing, internal chain transitivity and α-limit sets

Good

Meddaugh

Mitchell

2020

Journal of Mathematical Analysis and Applications

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Special α-limit points and unilateral γ-limit points for graph maps

Abstract: Let G be a graph and f : G → G be a continuous map. Denote by P (f ), R(f ), SA(f ) and U Γ(f ) the sets of periodic points, recurrent points, special α-limit points and unilateral γ-limit points of f , respectively. In this paper, we show that

Cited by 15 publications

References 15 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

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

Shadowing, internal chain transitivity and α-limit sets

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