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 $\partial\BB_Λ$ that is the infinite path space of the stationary $k$-Bratteli diagram $\BB_Λ$, where $\Lambda$ is a finite strongly connected $k$-graph. The Dirichlet form which we are interested in is induced by an even spectral triple $(C_{\operatorname{Lip}}(\PB_Λ), π_ϕ, \mathcal{H}, D, Γ)$ and is given by \[ Q_s(f,g)=\frac{1}{2} \int_Ξ \operatorname{Tr}\big(\vert D\vert^{-s} [D,π_ϕ(f)]^{\ast} [D,π_ϕ(g)] \big) \, dν(ϕ), \] where $\Xi$ is the space of choice functions on $\partial \BB_Λ\times \partial \BB_Λ$. There are two ultrametrics, $d^{(s)}$ and $d_{w_{\delta }}$, on $\partial \BB_Λ$ which make the infinite path space $\PB_Λ$ an ultrametric Cantor set. The former $d^{(s)}$ is associated to the eigenvalues of Laplace-Beltrami operator ${\Delta }_s$ associated to $Q_s$, and the latter $d_{w_{\delta }}$ is associated to a weight function $w_{\delta }$ on $\BB_Λ$, where $\delta \in (0,1)$. We show that the Perron-Frobenius measure $\mu$ on $\partial \BB_Λ$ has the volume doubling property with respect to both $d^{(s)}$ and $d_{w_{\delta }}$ 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 ${\mathcal{Q}}_{J_s,\mu }$ which is associated to a jump kernel $J_s$ and the measure $\mu$ on $\partial \BB_Λ$, and we investigate the asymptotic behavior and moments of displacements of the process.

to appear at J. Aust. Math. Soc