超度量Cantor集上的非交换Riemannian Geometry与扩散
Noncommutative Riemannian Geometry and Diffusion on Ultrametric Cantor Sets
摘要：
利用非交换几何的工具，对超度量Cantor集（C，d）描述了一个类似的黎曼几何.与（C，d）相关联的是一个加权根树，它的Michon树。这棵树允许定义一个家庭的频谱三元组给予康托集的结构的非交换黎曼流形。谱三元组的家庭是由空间的选择功能，这是被证明是一个黎曼流形的球丛的模拟索引。来自狄拉克算子D的康纳斯度规则允许恢复C上的度规。相应的zeta函数的收敛横坐标等于（C，d）的上盒维数。取这个奇点处的留数导致C上的一个规范概率测度的定义，它在某些情况下与豪斯多夫测度一致。这个度量反过来又导出了一个关于选择空间的度量。给定一个选择，D与Lipschitz连续函数的交换子可以解释为一个方向导数.通过对所有选择进行积分，这导致了拉普拉斯-贝尔特拉米算子的类似物的定义。该算子具有紧的预解式，并生成一个马尔可夫半群，该半群在C上起布朗运动的作用。这种构造适用于最简单的情况，三元康托集。
英文摘要：
An analogue of the Riemannian Geometry for an ultrametric Cantor set (C, d) is described using the tools of Noncommutative Geometry. Associated with (C, d) is a weighted rooted tree, its Michon tree. This tree allows to define a family of spectral triples giving the Cantor set the structure of a noncommutative Riemannian manifold. The family of spectral triples is indexed by the space of choice functions which is shown to be the analogue of the sphere bundle of a Riemannian manifold. The Connes metric coming from the Dirac operator D then allows to recover the metric on C. The corresponding zeta function is shown to have abscissa of convergence equal to the upper box dimension of (C, d). Taking the residue at this singularity leads to the definition of a canonical probability measure on C which in certain cases coincides with the Hausdorff measure. This measure in turns induces a measure on the space of choices. Given a choice, the commutator of D with a Lipschitz continuous function can be intepreted as a directional derivative. By integrating over all choices, this leads to the definition of an analogue of the Laplace-Beltrami operator. This operator has compact resolvent and generates a Markov semigroup which plays the role of a Brownian motion on C. This construction is applied to the simplest case, the triadic Cantor set.
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