Self-similar Energies On Finitely Ramified Fractals

Analysis of Fractals began to take shape as a mathematical field in the late 1980s. Traditionally, the focus of analysis has been on finitely ramified fractals — those where copies intersect at only finitely many points. To date, a comprehensive theory for infinitely ramified fractals remains elusive.This monograph outlines the theory of self-similar energies on finitely ramified self-similar fractals. A self-similar fractal is a non-empty, compact subset ℱ of a metric space (X, d) that satisfies ℱ = kSi=1ψi(ℱ) where ψi are a finite number of contractive similarities. Using these self-similar energies, one can construct Laplacians, harmonic functions, Brownian motion, and differential equations specific to these fractals.On finitely ramified fractals, self-similar energies are derived from eigenforms — quadratic forms that are eigenvectors of a special nonlinear operator within a finite-dimensional function space. The monograph also explores conditions for the existence and uniqueness of these self-similar energies and addresses related problems. For certain cases, complete solutions are provided.