Topological entropy of a graph map

Sun, Tai Xiang

doi:10.1007/s10114-017-7236-6

Cited by 4 publications

(4 citation statements)

References 9 publications

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“…Clearly, the set of pair of points a, b satisfying the above condition is countable (see also [20,Proposition 4.11]). Since, by [27], ω f (x) is uncountable, we conclude that infinitely many points of I ∩ ω f (x) satisfies condition (iii), and the proof finishes.…”

Section: The Sequence Topological Entropy Of Tree Mapsmentioning

confidence: 60%

Topological sequence entropy and topological dynamics of tree maps

Daghar¹,

Cánovas²

2022

Preprint

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Section: The Sequence Topological Entropy Of Tree Mapsmentioning

confidence: 60%

Topological sequence entropy and topological dynamics of tree maps

Daghar¹,

Cánovas²

2022

Preprint

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“…Hero used sα-limit sets to characterize the attracting centre Λ 2 ( f ) of a continuous interval map f , and some work has been done to extend his results to trees and graphs [15,26,27]. We conjecture that for graph maps, the β-limit sets can be used to characterize the Birkhoff centre Rec( f ) as follows:…”

Section: Problem 52mentioning

confidence: 99%

“…The connection to backward limit sets was made in 1992 by Hero, who defined special α-limit sets (sα-limit sets, for short) and showed that a point x belongs to the attracting centre of a continuous interval map if and only if it belongs to its own sα-limit set [15]. Partial generalizations of this result to graph maps and some dendrite maps have appeared since then [26][27][28].…”

Section: Introductionmentioning

confidence: 99%

On backward attractors of interval maps*

Hantáková

Roth

2021

Nonlinearity

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Special α-limit sets (sα-limit sets) combine together all accumulation points of all backward orbit branches of a point x under a noninvertible map. The most important question about them is whether or not they are closed. We challenge the notion of sα-limit sets as backward attractors for interval maps by showing that they need not be closed. This disproves a conjecture by Kolyada, Misiurewicz, and Snoha. We give a criterion in terms of Xiong’s attracting centre that completely characterizes which interval maps have all sα-limit sets closed, and we show that our criterion is satisfied in the piecewise monotone case. We apply Blokh’s models of solenoidal and basic ω-limit sets to solve four additional conjectures by Kolyada, Misiurewicz, and Snoha relating topological properties of sα-limit sets to the dynamics within them. For example, we show that the isolated points in a sα-limit set of an interval map are always periodic, the non-degenerate components are the union of one or two transitive cycles of intervals, and the rest of the sα-limit set is nowhere dense. Moreover, we show that sα-limit sets in the interval are always both F σ and G δ . Finally, since sα-limit sets need not be closed, we propose a new notion of β-limit sets to serve as backward attractors. The β-limit set of x is the smallest closed set to which all backward orbit branches of x converge, and it coincides with the closure of the sα-limit set. At the end of the paper we suggest several new problems about backward attractors.

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“…Recently Sun [40] generalised some equivalent conditions for an interval map has positive topological entropy to graph map.…”

Section: Conclusion and Recent Developmentmentioning

confidence: 99%

A Note on the Notions of Topological Entropy

Sang

Salleh

2018

EJMS

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Topological entropy is used to determine the complexity of a dynamical system. This paper aims to serve as a stepping stone for the study of topological entropy. We review the notions of topological entropy, give an overview on the relation between the notions and fundamental properties of topological entropy. Besides, we cover the topological entropy of the induced hyperspaces and its connection with the original systems. We also provide a summary on the latest research topic related with topological entropy.

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Topological entropy of a graph map

Cited by 4 publications

References 9 publications

Topological sequence entropy and topological dynamics of tree maps

Topological sequence entropy and topological dynamics of tree maps

On backward attractors of interval maps*

A Note on the Notions of Topological Entropy

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