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.