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

Abstract

The aim of this paper is to study the heat kernel and jump kernel of the Dirichlet form associated to ultrametric Cantor sets ∂B Λ that is the infinite path space of the stationary k-Bratteli diagram B Λ , where Λ is a finite strongly connected k-graph. The Dirichlet form which we are interested in is induced by an even spectral triple (C Lip (∂B Λ ), π φ , H, D, Γ) and is given bywhere Ξ is the space of choice functions on ∂B Λ × ∂B Λ . There are two ultrametrics, d (s) and d w δ , on ∂B Λ which make the infinite path space ∂B Λ an ultrametric Cantor set. The former d (s) is associated to the eigenvalues of Laplace-Beltrami operator ∆ s associated to Q s , and the latter d w δ is associated to a weight function w δ on B Λ , where δ ∈ (0, 1). We show that the Perron-Frobenius measure µ on ∂B Λ has the volume doubling property with respect to both d (s) and d w δ and we study the asymptotic behaviors of the heat kernel associated to Q s . Moreover, we show that the Dirichlet form Q s coincides with a Dirichlet form Q Js,µ which is associated to a jump kernel J s and the measure µ on ∂B Λ , and we investigate the asymptotic behavior and moments of displacements of the process.2010 Mathematics Subject Classification: 46L87, 60J75.