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HP Labs Technical Reports
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Finitely Ramified Graph Directed Fractals, Spectral Asympototics and the Multidimensional Renewal Theorem
Hambly, B.M.; Nyberg, S.O.G.
HPL-BRIMS-1999-12
Keyword(s):fractals; Laplace operators; spectral asymptotics;
renewal theory; Dirichlet forms; spectral dimension
Abstract:We consider the class of graph directed constructions
which are connected and have the property of finite
ramification. By assuming the existence of a fixed
point for a certain renormalization map, it is
possible to construct a Laplace operator on fractals
in this class via their Dirichlet forms. Within this
framework we are able to consider some fractals which
have infinitely ramified generators. Our main aim is
to consider the eigenvalues of the Laplace operator
and provide a formula for the spectral dimension, the
exponent determining the power law scaling in the
eigenvalue counting function, and establish generic
constancy for the counting function asymptotics. In
order to do this we prove an extension of the
multidimensional renewal theorem. As a result we show
that it is possible for the eigenvalue counting
function for fractals to require a logarithmic
correction to the usual power law growth. Notes:
S.O.G. Nyberg, Department of Mathematics and
Statistics, University of Edinburgh, Mayfield Road,
Edinburgh, EH9 3JZ, UK
34 Pages
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