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

Abstract

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 $\hat{\mathbb{C}}=\mathbb{C}\cup \{\infty \}$), 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.

arXiv comment22 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")

References in corpus (2)

• Energy measures and indices of Dirichlet forms, with applications to derivatives on some fractals
• The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues: A survey of the ergodic-theoretic aspects

Cited by in corpus (1)

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