MSI Weekly Bulletin - Week starting Monday 12 February, 2007

In the second part of the talk, we present how renewal theory can be applied in order to get a refinement of the spectral asymptotics of (measure geometric) second order differential operators d/dm d/dn. In the so called "lattice" or "arithemetic" case, high symmetry and self-similarity of the fractal and the measures may create oscillations in the asymptotic behaviour of the eigenvalue counting function. In some special cases, we obtain an exact renormalization formula for the Neumann eigenvalues by the method of Prufer angles.

Many physical phenomena proceed in or on irregular objects which are often modeled by fractal sets. Using the model case of the Sierpinski gasket, the notions of Hausdorff, spectral and walk dimension are introduced in a survey style. These "characteristic" numbers of the fractal are essential for the Einstein relation, expressing the interaction of geometric, analytic and stochastic aspects of a set. Keywords: fractals, self-similarity, Hausdorff dimension, Dirichlet form, Laplacian, spectral dimension, (strong) Markovian process, walk dimension, Einstain relation.