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VOL. 87 | 2021 The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues: A survey of the analytic aspects
Naotaka Kajino

Editor(s) Yuzuru Inahama, Hirofumi Osada, Tomoyuki Shirai

Abstract

This article surveys the analytic aspects of the author's recent studies on the construction and analysis of a “geometrically canonicalLaplacian on circle packing fractals invariant with respect to certain Kleinian groups (i.e., discrete groups of Möbius transformations on the Riemann sphere $\widehat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$), including the classical Apollonian gasket and some round Sierpiński 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.

Information

Published: 1 January 2021
First available in Project Euclid: 20 January 2022

Digital Object Identifier: 10.2969/aspm/08710293

Subjects:
Primary: 28A80 , 35P20 , 53C23
Secondary: 31C25 , 37B10 , 60J35

Keywords: Apollonian gasket , Dirichlet forms , Kleinian groups , Laplacian , round Sierpiński carpets , Weyl's eigenvalue asymptotics

Rights: Copyright © 2021 Mathematical Society of Japan

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