Noncommutative Riemannian Geometry and Diffusion on Ultrametric Cantor Sets

Abstract

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.