Li-Yorke chaos for ultragraph shift spaces

TL;DR

This paper investigates Li-Yorke chaos in ultragraph shift spaces, characterizing when such chaos occurs and showing the existence of compact scrambled sets, advancing understanding in dynamical systems linked to C*-algebra theory.

Contribution

It introduces metrics for ultragraph shift spaces and characterizes conditions for Li-Yorke chaos, including the existence of compact scrambled sets.

Findings

Characterization of ultragraph shift spaces with Li-Yorke chaos

Existence of perfect and scrambled sets in chaotic ultragraph shift spaces

Compact scrambled sets can be constructed in ultragraph shift spaces

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