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

Part of: Selfadjoint operator algebras Markov processes

Published online by Cambridge University Press:  08 January 2020

JAESEONG HEO Open the ORCID record for JAESEONG HEO [Opens in a new window] ,
SOORAN KANG Open the ORCID record for SOORAN KANG [Opens in a new window]  and
YONGDO LIM
JAESEONG HEO
Affiliation:
Department of Mathematics, Research Institute for Natural Sciences,Hanyang University, Seoul, 04763, Republic of Korea e-mail: hjs@hanyang.ac.kr
SOORAN KANG*
Affiliation:
College of General Education,Chung-Ang University, Seoul, 06974, Republic of Korea
YONGDO LIM
Affiliation:
Department of Mathematics,Sungkyunkwan University, Suwon, 16419, Republic of Korea e-mail: ylim@skku.edu
*
e-mail: sooran09@cau.ac.kr
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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.

Keywords

Dirichlet formsk-graphs and k-Bratteli diagramsheat kernelsasymptotic behaviorsultrametric Cantor sets

MSC classification

Primary: 46L87: Noncommutative differential geometry
Secondary: 60J75: Jump processes
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 110 , Issue 2 , April 2021 , pp. 194 - 219
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by A. Sims

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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