Transition probabilities are calculated which make the construction of diffusions on finitely ramified fractals straightforward. In contrast to former approaches using Brouwer's Fixed Point Theorem, we consider an approximation procedure based on the iteration of a nonlinear mapL. Physically, this is done by ‘coarse-graining-renormalisation of finite electric resistor networks’. Mathematically, it is a convergence problem for quotients of Dirichlet forms on finite graphs. These graphs approximate finitely ramified fractals. The basic tool is a contraction theorem for the renormalisation mapLwhich allows the use of known results about nested fractals for non-nested (p.c.f. self-similar) ones. In general, the above contraction is not strict because several linear independent fixed points occur.
The Friedland–Hayman inequality and Caffarelli’s contraction theorem
