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

This article surveys the analytic aspects of the author's recent studies on the construction and analysis of a "geometrically canonical" Laplacian on circle packing fractals invariant with respect to certain Kleinian groups (i.e., discrete groups of M\"{o}bius transformations on...

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Bibliographic Details
Published in:arXiv.org 2021-05
Main Author: Kajino, Naotaka
Format: Article
Language:English
Subjects:
Asymptotic properties
Carpets
Eigenvalues
Fractals
Riemann manifold
Online Access:Get full text
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Summary:This article surveys the analytic aspects of the author's recent studies on the construction and analysis of a "geometrically canonical" Laplacian on circle packing fractals invariant with respect to certain Kleinian groups (i.e., discrete groups of M\"{o}bius transformations on the Riemann sphere \(\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}\)), including the classical Apollonian gasket and some round Sierpi\'{n}ski carpets. The main result on Weyl's asymptotics for its eigenvalues is of the same form as that by Oh and Shah [Invent. Math. 187 (2012), 1--35, Theorem 1.4] on the asymptotic distribution of the circles in a very large class of such fractals.
ISSN:2331-8422