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 [Mic]. This tree allows to dene a family of spectral triples (CLip(C);H;D) using the ` 2-space of its vertices, giving the Cantor set the structure of a noncommutative Riemannian manifold. Here CLip(C) denotes the space of Lipschitz continuous functions on (C; d). 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 -function is shown to have abscissa of convergence, s0, equal to the upper box dimension of (C; d). Taking the residue at this singularity leads to the denition of a canonical probability measure on C which in certain cases coincides with the Hausdor measure at dimension s0. 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 denition 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 o