置换镶嵌的横截拉普拉斯算子
Transverse Laplacians for Substitution Tilings
摘要：
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。
英文摘要：
Pearson and Bellissard recently built a spectral triple - the data of Riemanian noncommutative geometry - for ultrametric Cantor sets. They derived a family of Laplace-Beltrami like operators on those sets. Motivated by the applications to specific examples, we revisit their work for the transversals of tiling spaces, which are particular self-similar Cantor sets. We use Bratteli diagrams to encode the self-similarity, and Cuntz-Krieger algebras to implement it. We show that the abscissa of convergence of the zeta-function of the spectral triple gives indications on the exponent of complexity of the tiling. We determine completely the spectrum of the Laplace-Beltrami operators, give an explicit method of calculation for their eigenvalues, compute their Weyl asymptotics, and a Seeley equivalent for their heat kernels.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
