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 partial derivative B-Lambda that is the infinite path space of the stationary k-Bratteli diagram B-Lambda, where Lambda is a finite strongly connected k-graph. The Dirichlet form which we are interested in is induced by an even spectral triple (C-Lip(partial derivative B-Lambda), pi(phi), H, D, Gamma) and is given by

Q(s)(f, g) = 1/2 integral(Xi) Tr(vertical bar D vertical bar(-s)[D, pi(phi) f)]* [D, pi(phi)(g)]) dv(phi),

where Xi is the space of choice functions on partial derivative B-Lambda x partial derivative B-Lambda. There are two ultrametrics, d((s)) and d(w delta), on partial derivative B-Lambda which make the infinite path space partial derivative B-Lambda an ultrametric Cantor set. The former d((s)) is associated to the eigenvalues of the Laplace-Beltrami operator Delta(s) associated to Q(s), and the latter d(w delta) is associated to a weight function w(delta) on B-Lambda, where delta is an element of 2 (0, 1). We show that the Perron-Frobenius measure mu on partial derivative B-Lambda has the volume-doubling property with respect to both d(s) and dw ffi and we study the asymptotic behavior of the heat kernel associated to Q(s). Moreover, we show that the Dirichlet form Qs coincides with a Dirichlet form QJ(s,mu) which is associated to a jump kernel J(s) and the measure mu on partial derivative B-Lambda, and we investigate the asymptotic behavior and moments of displacements of the process.