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The Laplacian on some self-conformal fractals and Weyl’s asymptotics for its eigenvalues: a survey of the analytic aspects. (English) Zbl 1486.35315

Inahama, Yuzuru (ed.) et al., Stochastic analysis, random fields and integrable probability – Fukuoka 2019. Proceedings of the 12th Mathematical Society of Japan, Seasonal Institute (MSJ-SI), Kyushu University, Japan, 31 July – 9 August 2019. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 87, 293-315 (2021).
Summary: 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 H. Oh and N. Shah [Invent. Math. 187, No. 1, 1–35 (2012; Zbl 1235.52033), Theorem 1.4] on the asymptotic distribution of the circles in a very large class of such fractals.
For the entire collection see [Zbl 1482.60003].

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
28A80 Fractals
31C25 Dirichlet forms
37B10 Symbolic dynamics
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
60J35 Transition functions, generators and resolvents

Citations:

Zbl 1235.52033
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Full Text: DOI arXiv

References:

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