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This short survey is aimed at sketching the ergodic-theoretic aspects of theauthor's recent studies on Weyl's eigenvalue asymptotics for a\emph{"geometrically canonical" Laplacian} defined by the author on someself-conformal circle packing fractals including the classical \emph{Apolloniangasket}. The main result being surveyed is obtained by applying Kesten'srenewal theorem [\emph{Ann.\ Probab.}\ \textbf{2} (1974), 355--386, Theorem 2]for functionals of Markov chains on general state spaces and provides analternative probabilistic proof of the result by Oh and Shah [\emph{Invent.\Math.}\ \textbf{187} (2012), 1--35, Corollary 1.8] on the asymptoticdistribution of the circles in the Apollonian gasket.

The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues: A survey of the ergodic-theoretic aspects