Continuous shift commuting maps between ultragraph shift spaces
Daniel Gonçalves, Marcelo Sobottka
备注：
14 pages
发表信息：
D. Gonçalves and M. Sobottka. Continuous shift commuting maps between ultragraph shift spaces. Disc. and Cont. Dynamic. Systems (2019), 39, 2, 1033-1048
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.
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We introduce the concept of a 1-coaligned $k$-graph and prove that the shift maps of a $k$-graph pairwise *-commute if and only if the $k$-graph is 1-coaligned. We then prove that for 2-graphs $Λ$ generated from basic data *-commuting shift maps is equivalent to a condition that implies that $C^*(Λ)$ is simple and purely infinite. We then consider full shift spaces and introduce a condition on a block map which ensures the associated sliding block code *-commutes with the shift.
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A countable family of $*$-commuting surjective, non-injective local homeomorphisms of a compact Hausdorff space $X$ gives rise to an action $θ$ of a countably generated, free abelian monoid $P$. For such a triple $(X,P,θ)$, which we call an irreversible $*$-commutative dynamical system, we construct a universal $C^*$-algebra $\mathcal{O}[X,P,θ]$. Within this setting we show that the following four conditions are equivalent: $(X,P,θ)$ is topologically free, $C(X) \subset \mathcal{O}[X,P,θ]$ has the ideal intersection property, the natural representation of $\mathcal{O}[X,P,θ]$ on $\ell^2(X)$ is…
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Let $f$ and $g$ be two quasiregular maps in $\mathbb{R}^d$ that are of transcendental type and also satisfy $f\circ g =g \circ f$. We show that if the fast escaping sets of those functions are contained in their respective Julia sets then those two functions must have the same Julia set. We also obtain the same conclusion about commuting quasimeromorphic functions with infinite backward orbit of infinity. Furthermore we show that permutable quasiregular functions of the form $f$ and $g=ϕ\circ f$, where $ϕ$ is a quasiconformal map, have the same Julia sets and that polynomial type quasiregular …
Recently Ott, Tomforde and Willis introduced a notion of one-sided shifts over infinite alphabets and proposed a definition for sliding block codes between such shift spaces. In this work we propose a more general definition for sliding block codes between Ott-Tomforde-Willis shift spaces and then we prove Curtis-Hedlund-Lyndon type theorems for them, finding sufficient and necessary conditions under which the class of the sliding block codes coincides with the class of continuous shift-commuting maps.
