基于 $AF$-代数的二维替换铺嵌空间的 $K$-理论

On the K-theory of C*-algebras for substitution tilings (a pedestrian version)

Under suitable conditions, a substitution tiling gives rise to a Smale space, from which three equivalence relations can be constructed, namely the stable, unstable, and asymptotic equivalence relations. We denote with $S$, $U$, and $A$ their corresponding $C^*$-algebras in the sense of Renault. In this article we show that the $K$-theories of $S$ and $U$ can be computed from the cohomology and homology of a single cochain complex with connecting maps for tilings of the line and of the plane. Moreover, we provide formulas to compute the $K$-theory for these three $C^*$-algebras. Furthermore, w…

Fractal spectral triples on Kellendonk's $C^*$-algebra of a substitution tiling

We introduce a new class of noncommutative spectral triples on Kellendonk's $C^*$-algebra associated with a nonperiodic substitution tiling. These spectral triples are constructed from fractal trees on tilings, which define a geodesic distance between any two tiles in the tiling. Since fractals typically have infinite Euclidean length, the geodesic distance is defined using Perron-Frobenius theory, and is self-similar with scaling factor given by the Perron-Frobenius eigenvalue. We show that each spectral triple is $θ$-summable, and respects the hierarchy of the substitution system. To elucida…

置换平铺C* 代数上的有限对称群作用

Finite symmetry group actions on substitution tiling C*-algebras

对于非周期置换镶嵌系统$(\p,ω)$的有限对称群$G$，证明了镶嵌C* 代数$\Aw$与$G$的交叉积具有实秩零、迹秩1和唯一迹，且其K-理论的阶由迹决定.证明了$G$在$\Aw$上的作用满足弱Rokhlin性质，并且只要$\Aw$的迹秩为零，它也满足迹Rokhlin性质.在证明后者的过程中，我们证明了$\Aw$是可生成的。我们还提供了一个链接之间的$\Aw$和AF代数Connes相关的彭罗斯镶嵌。

平铺群胚与Bratteli图II：轨道等价关系的结构

Tiling groupoids and Bratteli diagrams II: structure of the orbit equivalence relation

在这第二篇文章中，我们研究R^d的替换镶嵌的情况。在图块上的替换引起在所有维度j=0，.，d-1。我们重建平铺的等价关系在一个纯粹的组合的方式使用AF关系给出的低维替换。我们定义一个Bratteli多重图B，它由Bratteli图B^j，j=0，.，d，所有这些替换。B^d中的无限路径集与瓦片的典范横截线Xi相同。任何这样的路径都有一个“边界”，它是B^j中的一组尾部，其中j小于或等于d，这对应于其相关平铺的边界的自然概念。我们定义了B上的尾等价关系R_B，即两条无限长路等价，如果它们的边界在B^j中尾等价，其中j小于或等于d.我们证明了R_B同胚于镶嵌的等价关系R_Xi。

K-theory of the Chair Tiling via AF-algebras

We compute the $K$-theory groups of the groupoid C$^\ast$-algebra of the chair tiling, using a new method. We use exact sequences of Putnam to compute these groups from the $K$-theory groups of the $AF$-algebras of the substitution and the induced lower dimensional substitutions on edges and vertices.

置换镶嵌的横截拉普拉斯算子

Transverse Laplacians for Substitution Tilings

皮尔森和贝利萨德最近建立了一个谱三元组-黎曼非交换几何的数据-超度量康托集。他们在这些集合上导出了一个类似拉普拉斯-贝尔特拉米算子的家族。出于对具体例子的应用，我们重新审视了他们的工作的平铺空间，这是特殊的自相似康托集的横截。我们使用Bratteli图编码的自相似性，和Cuntz-Krieger代数来实现它。我们表明，收敛的zeta函数的谱三元组的横坐标给出指示的指数上的复杂性的平铺。我们完全确定了Laplace-Beltrami算子的谱，给出了其特征值的一个明确的计算方法，计算了它们的Weyl渐近性，以及它们的热核的一个Seeley等价。