Ultragraph shift spaces and chaos

Abstract

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,

Li-Yorke Chaos for ultragraph shift spaces

“…It is shown, see [25,Proposition 5.4] for example, that (X, σ) is a Deaconu-Renault system and the infinite sequences are dense in X. Moreover, a family of metrics for the topology in X is described in [31], see also [16,17]. We summarize this in the proposition below.…”

Ultragraph algebras via labelled graph groupoids, with applications to generalized uniqueness theorems

“…[10] where the description of the groupoid in the graph case is given). The unit space of this groupoid (also known as the boundary path space), with the associated shift map, is proposed to be a version of a shift of finite type over an infinite alphabet in [27,46], and the dynamics of these shifts is studied in [29,30,31], among other papers. In this subsection, we show that the unit space of the groupoid defined in [46] agrees with the tight spectrum T associated with the ultragraph.…”

“…Ultragraphs were originally defined in [47], as an unifying object to study Exel-Laca and graph C*-algebras. Since then, their study has intertwined Dynamics, Algebra and Analysis in ways that each area benefits from the other, see for example [26,29,30,31,46] where ultragraphs are used to study shift spaces over infinite alphabets, [15,46] where KMS states associated to ultragraph C*-algebras are described, [37] where purely infinite ultragraph C*-algebras are determined, [16] where topological full groups associated to ultragraph groupoids are shown to be isomorphism invariants, [35] where ultragraph Leavitt path algebras are introduced, [28] where irreducible representations of ultragraph Leavitt path algebras are characterized, among many other developments.…”

KMS states and continuous orbit equivalence for ultragraph shift spaces with sinks

“…Very recently, building from the ideas in [33,38], the authors of [17] introduce the notion of an ultragraph shift space, which is a generalization (to the infinite alphabet) of shifts of finite type (SFT are among the most important shifts in symbolic dynamics, see for example [30]). These shifts have interesting dynamics, as their chaotic behavior coincide with the chaotic behavior of shifts of finite type over finite alphabets (see [24,25]), and Curtis-Hedlund-Lyndon type results can be proved (see [21]). Furthermore, ultragraph shift spaces are showed in [17] to have a strong connection with C*-algebras: if two ultragraphs have associated shift spaces that are conjugate, via a conjugacy that preserves length, then the associated ultragraph C*-algebras are isomorphic.…”