Dirichlet Forms and Ultrametric Cantor Sets Associated to Higher-Rank Graphs

Abstract

The aim of this paper is to study the heat kernel and the jump kernel of the Dirichlet form associated to the ultrametric Cantor set that is the infinite path space of the stationary -Bratteli diagram, where is a finite strongly connected -graph. The Dirichlet form which we are interested in is induced by an even spectral triple and is given by where is the space of choice functions on. There are two ultrametrics, and, on which make the infinite path space an ultrametric Cantor set. The former is associated to the eigenvalues of the Laplace-Beltrami operator associated to, and the latter is associated to a weight function on, where. We show that the Perron-Frobenius measure on has the volume-doubling property with respect to both and and we study the asymptotic behavior of the heat kernel associated to. Moreover, we show that the Dirichlet form coincides with a Dirichlet form which is associated to a jump kernel and the measure on, and we investigate the asymptotic behavior and moments of displacements of the process. © 2020 Australian Mathematical Publishing Association Inc.