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arXiv · 2001.11354

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

Naotaka Kajino
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Abstract

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.}\ \textbf{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.}\ \textbf{187} (2012), 1--35, Corollary 1.8] on the asymptotic distribution of the circles in the Apollonian gasket.

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Naotaka Kajino. 2020-01-29. The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues: A survey of the ergodic-theoretic aspects. https://arxiv.org/abs/2001.11354

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