The Laplacian on some finitely ramified self-conformal circle packing fractals and Weyl’s asymptotics for its eigenvalues
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
the space of “shapes of cells’’ resulting from a suitable cellular
of the fractal.