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

This short survey is aimed at sketching the ergodic-theoretic aspects of the author’s recent studies on Weyl’s eigenvalue asymptotics for a \emph{“geometrically canonical” Laplacian} defined by the author on some self-conformal circle packing fractals including the classical \emph{Apollonian gasket}. The main result being surveyed is obtained by applying Kesten’s renewal theorem [\emph{Ann.\ Probab.}\ $\mathbf{\text{2}}$ (1974), 355–386, Theorem 2] for functionals of Markov chains on general state spaces and provides an alternative probabilistic proof of the result by Oh and Shah [\emph{Invent.
Math.}\ $\mathbf{\text{187}}$ (2012), 1–35, Corollary 1.8] on the asymptotic distribution of the circles in the Apollonian gasket.