arXiv2019
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 $Λ$ 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 $Ξ$ is the space of choice functions on $\partial \BB_Λ\times \partial \BB_Λ$. There are two ultrametrics, $d^{(s)}$ and $d_{w_δ}$, 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 $Δ_s$ associated to $Q_s$, and the latter $d_{w_δ}$ is associated to a weight function $w_δ$ on $\BB_Λ$, where $δ\in (0,1)$. We show that the Perron-Frobenius measure $μ$ on $\partial \BB_Λ$ 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 $\mathcal{Q}_{J_s, μ}$ which is associated to a jump kernel $J_s$ and the measure $μ$ on $\partial \BB_Λ$, and we investigate the asymptotic behavior and moments of displacements of the process.