Two-Sided Shift Spaces Over Infinite Alphabets

Sobottka, Marcelo; Starling, Charles

doi:10.1017/s1446788717000039

Cited by 16 publications

(38 citation statements)

References 31 publications

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“…Following the same outline of the proof of Proposition 3.6 in [15], one can prove that: Now that we have a good understanding of finitely defined sets we can define generalized sliding block codes.…”

Section: Generalized Sliding Block Codesmentioning

confidence: 94%

“…We remark that in the context of shift spaces with the product topology pseudo cylinders are equivalent to cylinders. On the other hand, for ultragraph shift spaces (and also in the context of the shift spaces studied in [14,15] and [21]), a pseudo cylinder is not necessarily an open set. However, as we will see in Proposition 3.12, a generalized cylinder, and its complement, can always be written as union of pseudo cylinders, that is, a generalized cylinder is a finitely defined set, accordingly to the following: Definition 3.9.…”

Section: Generalized Sliding Block Codesmentioning

confidence: 99%

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Continuous shift commuting maps between ultragraph shift spaces

Sobottka¹

2019

Discrete &Amp; Continuous Dynamical Systems - A

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Recently a generalization of shifts of finite type to the infinite alphabet case was proposed, in connection with the theory of ultragraph C*-algebras. In this work we characterize the class of continuous shift commuting maps between these spaces. In particular, we prove a Curtis-Hedlund-Lyndon type theorem and use it to completely characterize continuous, shift commuting, length preserving maps in terms of generalized sliding block codes.MSC 2010: 37B10, 54H20, 37B15

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Section: Generalized Sliding Block Codesmentioning

confidence: 94%

Section: Generalized Sliding Block Codesmentioning

confidence: 99%

Continuous shift commuting maps between ultragraph shift spaces

Sobottka¹

2019

Discrete &Amp; Continuous Dynamical Systems - A

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“…In graph C*-algebra theory it is usual to not define the shift map on elements of length zero (see for example [4]), but in symbolic dynamics this is often the case (see [18,19,25]). We therefore have chosen to define σ in all elements of X. is a homeomorphism.…”

Section: Remark 315mentioning

confidence: 99%

Infinite Alphabet Edge Shift Spaces via Ultragraphs and Their C*-Algebras

Royer

2017

International Mathematics Research Notices

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We define a notion of (one-sided) edge shift spaces associated to ultragraphs. In the finite case our notion coincides with the edge shift space of a graph. In general, we show that our space is metrizable and has a countable basis of clopen sets. We show that for a large class of ultragraphs the basis elements of the topology are compact.We examine shift morphisms between these shift spaces, and, for the locally compact case, show that if two (possibly infinite) ultragraphs have edge shifts that are conjugate, via a conjugacy that preserves length, then the associated ultragraph C*-algebras are isomorphic. To prove this last result we realize the relevant ultragraph C*-algebras as partial crossed products.MSC 2010: 37B10, 47L65, 54H20

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“…Even more, as g(J) and g(H) agree in all entries i ∈ ℓ≥0 E ℓ , the sentence (14) above is equivalent to:…”

Section: Chaos On Ultragraph Shift Spacesmentioning

confidence: 99%

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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Two-Sided Shift Spaces Over Infinite Alphabets

Cited by 16 publications

References 31 publications

Continuous shift commuting maps between ultragraph shift spaces

Continuous shift commuting maps between ultragraph shift spaces

Infinite Alphabet Edge Shift Spaces via Ultragraphs and Their C*-Algebras

Ultragraph shift spaces and chaos

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