Shadowing, finite order shifts and ultrametric spaces

Darji, Udayan B.; Sobottka, Marcelo

doi:10.1016/j.aim.2021.107760

Cited by 11 publications

(5 citation statements)

References 22 publications

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“…In the infinite alphabet case, with the product topology, a similar result holds, i.e. a shift has the shadowing property if and only if it is a shift of finite order (a generalization of a shift of finite type), see [8,21]. In the context of the Deaconu-Renault system associated with a graph, we provide an example of a graph with only two vertices in which the associated system does not possess the shadowing property.…”

Section: Introductionmentioning

confidence: 72%

“…Next we show that, for Deaconu-Renault systems, it is enough to verify shadowing in a dense subset. This is analogous to [8,Theorem 2.1.7], but we do not require uniform continuity of the map. Proposition 2.13.…”

Section: Shadowing For Local Homeomorphismsmentioning

confidence: 99%

“…For instance, in [27], it is proven that finite shadowing and infinite shadowing are equivalent for compact spaces. However, this equivalence does not hold in the general theory of shadowing for non-compact spaces (see [8]), nor does it hold for Deaconu-Renault systems. Additionally, other properties of the shadowing property need to be proven or verified.…”

Section: Introductionmentioning

confidence: 99%

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Li-Yorke Chaos for ultragraph shift spaces

Uggioni¹

2020

Discrete &Amp; Continuous Dynamical Systems - A

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Recently, in connection with C*-algebra theory, the first author and Danilo Royer introduced ultragraph shift spaces. In this paper we define a family of metrics for the topology in such spaces, and use these metrics to study the existence of chaos in the shift. In particular we characterize all ultragraph shift spaces that have Li-Yorke chaos (an uncountable scrambled set), and prove that such property implies the existence of a perfect and scrambled set in the ultragraph shift space. Furthermore, this scrambled set can be chosen compact, what is not the case for a labelled edge shift (with the product topology) of an infinite graph.MSC 2010: 37B10, 37B20, 37D40, 54H20,We begin with some standard definitions regarding ultragraphs, as introduced in [16] and [23]. After this we recall the definition of an ultragraph shift space X, as given in [8], and then define a family of metrics for the topology in X. BackgroundDefinition 2.1. An ultragraph is a quadruple G = (G 0 , G 1 , r, s) consisting of two countable sets G 0 , G 1 , a map s : G 1 → G 0 , and a map r : G 1 → P (G 0 ) \ {∅}, where P (G 0 ) stands for the power set of G 0 .

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Section: Introductionmentioning

confidence: 72%

Section: Shadowing For Local Homeomorphismsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Li-Yorke Chaos for ultragraph shift spaces

Uggioni¹

2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…Let E = (E 0 , E 1 , r, s) be the disconnected graph consisting of the above graph and a rose of three petals, see the picture below. Dynamical properties and the (Gurevich) topological entropy of the associated Markov shift space (the usual shift space equipped with the product topology, see [2]) are studied in [26,32].…”

Section: The Entropy Of a Row-finite Graphmentioning

confidence: 99%

“…23 we computed the entropy of the top subgraph, which is log(2). The entropy of the subgraph generated by {g 1 , g 2 , g 3 } and {u} is log(3).…”

mentioning

confidence: 99%

Entropy of local homeomorphisms with applications to infinite alphabet shift spaces

Royer¹,

Tasca²

2023

Preprint

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In this paper, we introduce topological entropy for dynamical systems generated by a single local homeomorphism (Deaconu-Renault systems). More precisely, we generalize Adler, Konheim, and McAndrew's definition of entropy via covers and Bowen's definition of entropy via separated sets. We propose a definition of factor map between Deaconu-Renault systems and show that entropy (via separated sets) always decreases under uniformly continuous factor maps. Since the variational principle does not hold in the full generality of our setting, we show that the proposed entropy via covers is a lower bound to the proposed entropy via separated sets. Finally, we compute entropy for infinite graphs (and ultragraphs) and compare it with the entropy of infinite graphs defined by Gurevich.

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Some Notes on the Classification of Shift Spaces: Shifts of Finite Type; Sofic Shifts; and Finitely Defined Shifts

Sobottka

2022

Bull Braz Math Soc, New Series

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Shadowing, finite order shifts and ultrametric spaces

Cited by 11 publications

References 22 publications

Li-Yorke Chaos for ultragraph shift spaces

Li-Yorke Chaos for ultragraph shift spaces

Entropy of local homeomorphisms with applications to infinite alphabet shift spaces

Some Notes on the Classification of Shift Spaces: Shifts of Finite Type; Sofic Shifts; and Finitely Defined Shifts

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