Continuous shift commuting maps between ultragraph shift spaces

Abstract

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

Blur shift spaces

“…In this section we present several results that characterize shift-commuting maps, with particular interest on the case where they are continuous or generalized sliding block codes. The results presented here hold for general blur shifts and they are expressed in the general framework, but their proofs share ideas with those given in [5] in the particular context of ultragraph shifts. We remark that, differently than in [6] where it was considered only Ott-Tomforde-Willis shift spaces over countable alphabets, or than in [5] where the alphabet was always countable, here we do not impose any restriction on the cardinality of the alphabet, which can be uncountable.…”

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.…”

Li-Yorke Chaos for ultragraph shift spaces

“…The definition in [8] relies on ultragraphs and the resulting shift space contains a countable basis of clopen subsets (which for ultragraphs that satisfy a mild condition turn to be compact-open subsets). Given the above setting, to understand, and compare, the dynamics of shifts over infinite alphabets is a relevant problem (note that Curtis-Hend-Lund type theorems for shifts over infinite alphabets were described in [11,13,21]). In this paper we study Li-Yorke Chaos for the ultragraph shift spaces defined in [8].…”