Li-Yorke Chaos for ultragraph shift spaces

Daniel Gonçalves; Bruno Brogni Uggioni

doi:10.3934/dcds.2020117

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2020, Volume 40, Issue 4: 2347-2365. Doi: 10.3934/dcds.2020117

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Li-Yorke Chaos for ultragraph shift spaces

Daniel Gonçalves^1, , and
Bruno Brogni Uggioni^2,

1.
UFSC – Departamento de Matemática, Florianópolis, SC 88040-900, Brazil
2.
IFRS – Campus Canoas, Canoas, RS 92412-240, Brazil

^* Corresponding author: Daniel Gonçalves
^* Corresponding author: Daniel Gonçalves

Received: July 2019

Published online: April 2020

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Abstract

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, which is not the case for a labelled edge shift (with the product topology) of an infinite graph.

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Mathematics Subject Classification: Primary: 37B10, 37B20; Secondary: 37D40, 54H20.

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