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

  • JAESEONG HEO (a1), SOORAN KANG (a2) and YONGDO LIM (a3)

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 $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ 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.

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The first author J.H. and the third author Y.L. were supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST), No. NRF-2015R1A3A2031159. The second author S.K. was supported by the Basic Science Research Program through a NRF grant funded by the Ministry of Education, No. NRF-2017R1D1A1B03034697.

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

  • JAESEONG HEO (a1), SOORAN KANG (a2) and YONGDO LIM (a3)

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