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 \emp...

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Bibliographic Details
Published in:arXiv.org 2020-01
Main Author: Kajino, Naotaka
Format: Article
Language:English
Subjects:
Asymptotic properties
Eigenvalues
Ergodic processes
Fractals
Markov chains
Theorems
Online Access:Get full text
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Summary: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.
ISSN:2331-8422