# Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets

### John Pearson

Georgia Institute of Technology, Atlanta### Jean Bellissard

Georgia Institute of Technology, Atlanta

## 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 [29]. This tree allows to define a family of spectral triples $(C_{Lip}(C),H,D)$ using the $ℓ_{2}$-space of its vertices, giving the Cantor set the structure of a noncommutative Riemannian manifold. Here $C_{Lip}(C)$ 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 family of these spectral triples allows to recover the metric on $C$. The corresponding $ζ$-function is shown to have abscissa of convergence, $s_{0}$, 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 at dimension $s_{0}$. This measure in turn induces a measure on the space of choices. Given a choice, the commutator of $D$ with a Lipschitz continuous function can be interpreted 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, where: (i) the spectrum and the eigenfunctions of the Laplace–Beltrami operator are computed, (ii) the Weyl asymptotic formula is shown to hold with the dimension $s_{0}$, (iii) the corresponding Markov process is shown to have an anomalous diffusion with $E(d(X_{t},X_{t+δt})_{2})≃δt ln(1/δt)$ as $δt↓0$.

## Cite this article

John Pearson, Jean Bellissard, Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets. J. Noncommut. Geom. 3 (2009), no. 3, pp. 447–480

DOI 10.4171/JNCG/43