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

Naotaka Kajino

备注：: 22 pages, 6 figures; the reference information has been modified and updated (to appear in The Proceedings of the 12th MSJ-SI "Stochastic Analysis, Random Fields and Integrable Probability")

英文摘要：

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 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.

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