Abstract

The purpose of this talk is to present the speaker’s recent research in progress on the construction of a ``canonical’’ Laplacian on finitely ramified circle packing fractals invariant with respect to a family of Moebius transformations and on Weyl’s asymptotics for its eigenvalues.

In the simplest case of the Apollonian gasket, the speaker has obtained an explicit expression of a certain canonical Dirichlet form in terms of the circle packing structure of the fractal. Our Laplacian on a general circle packing fractal is constructed by adopting the same kind of expression as the definition of a (seemingly canonical) strongly local Dirichlet form. Weyl’s eigenvalue asymptotics for this Laplacian has been also established in some important examples including the Apollonian gasket, and the proof of this result heavily relies on ergodic-theoretic analysis of a Markov chain on the space of ``shapes of cells’’ resulting from a suitable cellular decomposition of the fractal.