Dirichlet spaces connected by points and application to diffusions on finitely ramified fractals
The author's results [Ann. Sci. Éc. Norm. Supér., IV. Sér. 30, No. 5, 605-673 (1997; Zbl 0924.60064)] are complemented by computational methods and explicit numerics for the renormalization map \(T\) on his finitely ramified self-similar fractals. Instead of the Dirichlet form the author uses the corresponding \(0\)-order energy integral (the bilinear form defined by the Green operator on measures). As a consequence \(T\) becomes linear on fractal trees and an eigenvector can be found by classical Perron-Frobenius theory. The representations of the group of gobal symmetries of a fractal is used to express \(T\) in terms of the eigenvalues of the Dirichlet form, provided the group is sufficiently large. This explains a technique used by \textit{M. T. Barlow} [in: Asymptotic problems in probability theory: stochastic models and diffusions on fractals. Pitman Res. Notes Math. Ser. 283, 131-157 (1993; Zbl 0791.60097)]. For the Lindstrøm snowflake the map \(T\) is calculated explicitly in this way.
scientific article; zbMATH DE number 1340237