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 [29]. This tree allows to define a family of spectral triples $({\mathcal{C}}_{\mathrm{Lip}}(C),\mathcal{H},D)$ using the ${\ell }^2$-space of its vertices, giving the Cantor set the structure of a noncommutative Riemannian manifold. Here ${\mathcal{C}}_{\mathrm{Lip}}(C)$ 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 $\zeta$-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 $\mathbb{E}(d(X_t,X_{t+\delta t}{)}^2)\simeq \delta t\ln (1/\delta t)$ as $\delta t\downarrow 0$.