| Abstract: |
| The purpose of this talk is to present the speaker`s recent results on
the construction of a ``canonical`` Laplacian on circle packing fractals
invariant under the action of certain Kleinian groups
(i.e., discrete groups of Moebius transformations)
and on the asymptotic behavior of its eigenvalues. In the simplest case of
the Apollonian gasket, the Laplacian was constructed by Teplyaev (2004) as
one with respect to which the coordinate functions on the gasket are harmonic,
and the author has recently proved its uniqueness and discovered an explicit
expression of it in terms of the circle packing structure of the gasket.
The expression of the Laplacian actually makes sense on general circle packing
fractals and defines (a candidate of) a ``canonical`` Laplacian on such fractals.
When the fractal is the limit (i.e., minimum invariant non-empty compact)
set of a certain class of Kleinian groups, some explicit combinatorial
structure of the fractal is known and makes it possible to prove Weyl`s
asymptotic formula for the eigenvalues of this Laplacian, which is of
the same form as the circle-counting asymptotic formula by Oh and Shah
[Invent. Math., 2012]. Its proof is based on a serious application
of Kesten`s renewal theorem [Ann. Probab., 1974]. |
|