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 Laplaceoperator on fractals in this class via their Dirichlet forms. Our main aim is to consider the eigenvalues of theLaplace operator and provide a formula for the spectral dimension, the exponent determining the power-law scaling inthe eigenvalue counting function, and establish generic constancy for the counting-function asymptotics. In order to dothis we prove an extension of the multidimensional renewal theorem. As a result we show that it is possible for theeigenvalue counting function for fractals to require a logarithmic correction to the usual power-law growth.
AMS 2000 Mathematics subject classification: Primary 35P20; 58J50. Secondary 28A80; 60K05; 31C25
