Li-Yorke chaos for ultragraph shift spaces

Uggioni, Bruno Brogni

doi:10.48550/arxiv.1806.07927

Cited by 3 publications

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“…In this paper we show that the 'combinatorial' condition that characterizes Li-Yorke chaos for ultragraph shift spaces also characterizes distributional chaos and the existence of a uncountable, closed, shift invariant subset that is Devaney chaotic. Furthermore, we prove that the distributional uncountable scrambled set can be chosen to be compact and perfect, but this set is not the same as the one we built in [7] (which we show is not distributional chaotic). In particular, our results show that ultragraph edge shift spaces behave like cocyclic shifts (which generalize sofic shifts), as the equivalence between Li-Yorke, Devaney, and DCi chaos, in the context of finite alphabet cocyclic shift spaces, was proved in [21].…”

Section: Introductionmentioning

confidence: 81%

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Ultragraph shift spaces and chaos

Uggioni¹

2020

Bulletin des Sciences Mathématiques

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Motivated by C*-algebra theory, ultragraph edge shift spaces generalize shifts of finite type to the infinite alphabet case. In this paper we study several notions of chaos for ultragraph shift spaces. More specifically, we show that Li-Yorke, Devaney and distributional chaos are equivalent conditions for ultragraph shift spaces, and characterize this condition in terms of a combinatorial property of the underlying ultragraph. Furthermore, we prove that such properties imply the existence of a compact, perfect set which is distributionally scrambled of type 1 in the ultragraph shift space (a result that is not known for a labelled edge shift (with the product topology) of an infinite graph).MSC 2010: 37B10, 37B20, 37D40, 54H20,

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Section: Introductionmentioning

confidence: 81%

“…Previously, see [7], we have studied Li-Yorke chaos associated to the ultragraph shift spaces defined in [8]. In particular we have showed that Li-Yorke chaoticity is linked to the existence of a vertex in the ultragraph that is the base of two distinct closed paths.…”

Section: Introductionmentioning

confidence: 99%

Ultragraph shift spaces and chaos

Uggioni¹

2020

Bulletin des Sciences Mathématiques

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“…This restriction makes it much simpler to deal with objects (such as algebras and topological spaces) associated to ultragraphs than with objects associated to labelled graphs, but still interesting properties of labelled graphs present themselves in ultragraphs. This is the case, for example, in the study of Li-Yorke chaos for shift spaces over infinite alphabets (see [32]). In fact, ultragraphs are key in the study of shift spaces over infinite alphabets, see [28,29,31].…”

Section: Introductionmentioning

confidence: 97%

Representations and the reduction theorem for ultragraph Leavitt path algebras

Royer¹

2019

Preprint

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In this paper we study representations of ultragraph Leavitt path algebras via branching systems and, using partial skew ring theory, prove the reduction theorem for these algebras. We apply the reduction theorem to show that ultragraph Leavitt path algebras are semiprime and to completely describe faithfulness of the representations arising from branching systems, in terms of the dynamics of the branching systems. Furthermore, we study permutative representations and provide a sufficient criteria for a permutative representation of an ultragraph Leavitt path algebra to be equivalent to a representation arising from a branching system. We apply this criteria to describe a class of ultragraphs for which every representation (satisfying a mild condition) is permutative and has a restriction that is equivalent to a representation arising from a branching system.

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“…Ultragraphs (and their algebras) are generalizations of graphs (and their algebras) with applications in Symbolic Dynamics, Operator Algebras and Algebra, see for example [9,15,24,25,26,27,28,29,30,33,34,36,39,40]. The Leavitt path algebra associated to an ultragraph was defined in [32],…”

Section: Introductionmentioning

confidence: 99%

“…It is therefore interesting to extend known results of Leavitt path algebra theory to ultragraph Leavitt path algebras. Furthermore, since ultragraph Leavitt path algebras are algebraic analogues of ultragraph C*algebras, which are well studied and play a key role in the study of infinite alphabet shift spaces (see [26,27,28,29,30]), it is important to deepen the understanding of these algebras.…”

Section: Introductionmentioning

confidence: 99%

Irreducible and permutative representations of ultragraph Leavitt path algebras

Royer

2019

Forum Mathematicum

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We completely characterize perfect, permutative, irreducible representations of an ultragraph Leavitt path algebra. For this we extend to ultragraph Leavitt path algebras Chen's construction of irreducible representations of Leavitt path algebras. We show that these representations can be built from branching system and characterize irreducible representations associated to perfect branching systems. Along the way we improve the characterization of faithfulness of Chen's irreducible representations.

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Li-Yorke chaos for ultragraph shift spaces

Cited by 3 publications

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Ultragraph shift spaces and chaos

Ultragraph shift spaces and chaos

Representations and the reduction theorem for ultragraph Leavitt path algebras

Irreducible and permutative representations of ultragraph Leavitt path algebras

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