Li-Yorke Chaos for 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.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 .

Ultragraph shift spaces and chaos

“…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].…”

“…In this section we quickly review the construction of ultragraph shift spaces, as introduced in [8], and the associated metrics in these spaces, as defined in [7]. Definition 2.1.…”

“…In [8] a topology with a basis of cylinder sets was defined for X, and in [7] the authors showed that this topology coincides with the topology given by a metric. This metric was obtained by listing the elements of p as p = {p 1 , p 2 , p 3 , .…”

“…We have now developed the necessary tools to show that if an ultragraph shift space X has a DCi pair (formed by infinite paths), for some i ∈ {1, 2, 3}, then the associated ultragraph has a vertex v that is the base of two different closed paths (this should be compared with the results in [7] regarding Li-Yorke chaoticity). More precisely we have the following proposition.…”

“…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.…”