The Laplacian on some self-conformal fractals and Weyl's asymptotics for its eigenvalues: A survey of the ergodic-theoretic aspects (Research on the Theory of Random Dynamical Systems and Fractal Geometry)

Abstract

This short survey is aimed at sketching the ergodic-theoretic aspects of the author's recent studies on Weyl's eigenvalue asymptotics for a "geometrically canonical" Laplacian defined by the author on some self-conformal circle packing fractals including the classical Apollonian gasket. The main result being surveyed is obtained by applying Kesten's renewal theorem [Ann. Probab. 2 (1974), 355- 386, Theorem 2] for functionals of Markov chains on general state spaces and provides an alternative probabilistic proof of the result by Oh and Shah [Invent. Math. 187 (2012), 1-35, Corollary 1.8] on the asymptotic distribution of the circles in the Apollonian gasket.