Dirichlet Spaces Connected by Points and Application to Diffusions on Finitely Ramified Fractals

摘要

We illustrate the results of [13] by giving some computation methods and some explicite computations of the renormalization map involved in the construction of a diffusion on a finitely ramified self-similar set. Using the inverse of the Dirichlet form (defined on the set of finite 0-energy measures) we show that in some cases the map is linear and given by a nonnegative matrice. We also use the representations of the group of symmetries of the fractal to compute T in the coordinates associated with the eigenvalues of the Dirichlet form. In this way we give the meaning in terms of representations of groups of the change of variable made in [1]. We explicitely compute T for the snowflake and we are able to give an approximated value of the fixed point, and so of the probabilities of transition of the diffusion.