DIRICHLET FORMS AND ULTRAMETRIC CANTOR SETS ASSOCIATED TO HIGHER-RANK GRAPHS

Abstract

AbstractThe aim of this paper is to study the heat kernel and the jump kernel of the Dirichlet form associated to the ultrametric Cantor set $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ that is the infinite path space of the stationary $k$-Bratteli diagram ${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$, where $\unicode[STIX]{x1D6EC}$ is a finite strongly connected $k$-graph. The Dirichlet form which we are interested in is induced by an even spectral triple $(C_{\operatorname{Lip}}(\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}),\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}},{\mathcal{H}},D,\unicode[STIX]{x1D6E4})$ and is given by $$\begin{eqnarray}Q_{s}(f,g)=\frac{1}{2}\int _{\unicode[STIX]{x1D6EF}}\operatorname{Tr}(|D|^{-s}[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(f)]^{\ast }[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(g)])\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D719}),\end{eqnarray}$$ where $\unicode[STIX]{x1D6EF}$ is the space of choice functions on $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}\times \unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$. There are two ultrametrics, $d^{(s)}$ and $d_{w_{\unicode[STIX]{x1D6FF}}}$, on $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ which make the infinite path space $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ an ultrametric Cantor set. The former $d^{(s)}$ is associated to the eigenvalues of the Laplace–Beltrami operator $\unicode[STIX]{x1D6E5}_{s}$ associated to $Q_{s}$, and the latter $d_{w_{\unicode[STIX]{x1D6FF}}}$ is associated to a weight function $w_{\unicode[STIX]{x1D6FF}}$ on ${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$, where $\unicode[STIX]{x1D6FF}\in (0,1)$. We show that the Perron–Frobenius measure $\unicode[STIX]{x1D707}$ on $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ has the volume-doubling property with respect to both $d^{(s)}$ and $d_{w_{\unicode[STIX]{x1D6FF}}}$ and we study the asymptotic behavior of the heat kernel associated to $Q_{s}$. Moreover, we show that the Dirichlet form $Q_{s}$ coincides with a Dirichlet form ${\mathcal{Q}}_{J_{s},\unicode[STIX]{x1D707}}$ which is associated to a jump kernel $J_{s}$ and the measure $\unicode[STIX]{x1D707}$ on $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$, and we investigate the asymptotic behavior and moments of displacements of the process.