Tiling Groupoids and Bratteli Diagrams II: Structure of the Orbit Equivalence Relation

In this second paper, we study the case of substitution tilings of \mathbb R^d{{\mathbb R}^d} . The substitution on tiles induces substitutions on the faces of the tiles of all dimensions j = 0, . . . , d − 1. We reconstruct the tiling’s equivalence relation in a purely combinatorial way using the AF-relations given by the lower dimensional substitutions. We define a Bratteli multi-diagram B{{\mathcal B}} which is made of the Bratteli diagrams B^j, j=0, ?d{{\mathcal B}^j, j=0, \ldots d} , of all those substitutions. The set of infinite paths in B^d{{\mathcal B}^d} is identified with the canonical transversal Ξ of the tiling. Any such path has a “border”, which is a set of tails in B^j{{\mathcal B}^j} for some j ≤ d, and this corresponds to a natural notion of border for its associated tiling. We define an étale equivalence relation R_B{{\mathcal R}_{\mathcal B}} on B{{\mathcal B}} by saying that two infinite paths are equivalent if they have borders which are tail equivalent in B^j{{\mathcal B}^j} for some j ≤ d. We show that R_B{{\mathcal R}_{\mathcal B}} is homeomorphic to the tiling’s equivalence relation R_X{{\mathcal R}_\Xi} .     

S-adic conjecture and Bratteli diagrams

In this Note we apply a substantial improvement of a result of S. Ferenczi on S-adic subshifts to give Bratteli–Vershik representations of these subshifts.     

Cantor aperiodic systems and Bratteli diagrams

We prove that every Cantor aperiodic system is homeomorphic to the Vershik map acting on the space of infinite paths of an ordered Bratteli diagram and give several corollaries of this result. To cite this article: K. Medynets, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Termal Groupoids

We investigate the factor of the groupoid of terms through the largest congruence with a given set among its blocks. The set is supposed to be closed for overterms.     

Two-Vector Spaces and Groupoids

This paper describes a relationship between essentially finite groupoids and two-vector spaces. In particular, we show to construct two-vector spaces of Vect-valued presheaves on such groupoids. We define two-linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation—a weak functor—from Span(FinGpd) (the bicategory of essentially finite groupoids and spans of groupoids) into 2Vect. In this paper we prove this and give the construction in detail.     

Orbifolds, Sheaves and Groupoids

We characterize orbifolds in terms of their sheaves, and show that orbifolds correspond exactly to a specific class of smooth groupoids. As an application, we construct fibered products of orbifolds and prove a change-of-base formula for sheaf cohomology.     

Homeomorphic measures on stationary Bratteli diagrams

We study the set S of ergodic probability Borel measures on stationary non-simple Bratteli diagrams which are invariant with respect to the tail equivalence relation R. Equivalently, the set S is formed by ergodic probability measures invariant with respect to aperiodic substitution dynamical systems. The paper is devoted to the classification of measures μ from S with respect to a homeomorphism. The properties of the clopen values set S(μ) are studied. It is shown that for every measure μ∈S there exists a subgroup G⊂R such that S(μ)=G∩[0,1]. A criterion of goodness is proved for such measures. Based on this result, the measures from S are classified up to a homeomorphism. We prove that for every good measure μ∈S there exist countably many measures {μi}_i∈N⊂S such that the measures μ and μi are homeomorphic but the tail equivalence relations on the corresponding Bratteli diagrams are not orbit equivalent.     

Diagram Groupoids and von Neumann Algebras

The main purpose of this paper is to study certain algebraic structures induced by directed graphs. We have studied graph groupoids, which are algebraic structures induced by given graphs. By defining a certain groupoid-homomorphism ?? on the graph groupoid ${\mathbb{G}}$ of a given graph G, we define the diagram of G by the image ${\delta(\mathbb{G})}$ of ??, equipped with the inherited binary operation on ${\mathbb{G}}$ . We study the fundamental properties of the diagram ${\delta(\mathbb{G})}$ , and compare them with those of ${\mathbb{G}}$ . Similar to Cho (Acta Appl Math 95:95?C134, 2007), we construct the groupoid von Neumann algebra ${\mathcal{M}_{G}=vN(\delta(\mathbb{G}))}$ , generated by ${\delta(\mathbb{G})}$ , and consider the operator algebraic properties of ${\mathcal{M}_{G}}$ . In particular, we show ${\mathcal{M}_{G}}$ is *-isomorphic to a von Neumann algebra generated by a family of idempotent operators and nilpotent operators, under suitable representations.     

辛群胚与泊松群胚作用的充要条件

研究了辛群胚与泊松群胚的作用.利用李群胚作用及相关性质,得到了李群胚作用成为辛群胚和泊松群胚作用的充要条件,推广了辛群胚和泊松群胚的性质,为辛群胚与泊松群胚理论的进一步研究起到了推动作用.     

Spotlight Tiling

This article introduces spotlight tiling, a type of covering which is similar to tiling. The distinguishing aspects of spotlight tiling are that the “tiles” have elastic size, and that the order of placement is significant. Spotlight tilings are decompositions, or coverings, and can be considered dynamic as compared to typical static tiling methods. A thorough examination of spotlight tilings of rectangles is presented, including the distribution of such tilings according to size, and how the directions of the spotlights themselves are distributed. The spotlight tilings of several other regions are studied, and suggest that further analysis of spotlight tilings will continue to yield elegant results and enumerations.     

A Class of Groupoids

A new class of left distributive groupoids and their relationship with homogeneous spaces are studied. Bibliography: 5 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 305, 2003, pp. 136–143.     

广义单侧剩余广群

对吴氏左（右）剩余广群，给出一种新而且简化的定义，并详细地讨论其基本性质。用其主要结论得到完备格上的所有无穷∨-分配强伪t-模与所有无穷∧-分配强蕴涵算子之间的关系。     

Groupoid的诱导表示

设Ｇ为第二可数局部紧有Ｈａａｒ系的Ｇｒｏｕｐｏｉｄ， Ｈ为子Ｇｒｏｕｐｏｉｄ闭于Ｇ，则可得Ｇｒｏｕｐｏｉｄ　Ｈ＼Ｇ２，我们证明了Ｃ＊（Ｈ）与Ｃ＊（Ｈ＼Ｇ２）是Ｍｏｒｉｔａ等价的，从而回答了［１］中的问题．利用此非本原双模及定义Ｃ＊（Ｇ）到Ｍ（Ｃ＊（Ｈ＼Ｇ２））的映射，得到了由Ｃ＊（Ｈ）到Ｃ＊（Ｇ）的诱导表示．特别在群丛情形，我们定义了Ｃ＊（Ｈ）→Ｍ（Ｃ＊（Ｇ））的映射，并具体得到了诱导表示的积分形式的表达式．     

Ordered Groupoids and Left Cancellative Categories

We generalise work by Leech and Loganathan to obtain a correspondence between left cancellative categories and ordered groupoids. In particular, we prove that up to equivalence every category of monics can be obtained from an ordered groupoid.