SELF-SIMILAR ENERGIES ON FINITELY RAMIFIED FRACTALS

This monograph delves into the theory of self-similar energies on finitely ramified self-similar fractals. 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.

Analysis on 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 in which copies intersect at only finitely many points. To date, a comprehensive theory for infinitely ramified fractals remains elusive.

Contents:

• About the Author
• Introduction
• Notation and Preliminaries
• Self-Similar Fractals
• Energy on the Gasket
• Quadratic Forms
• Fractal Triples
• Energies on Fractal Triples
• Energies on a Fractal
• Existence of w-eigenforms
• Uniqueness
• Convergence of the Iterates of Renormalization Operator
• Existence with Suitable Weights
• Existence at a Suitable Level
• Energy Forms on Fractals as Limits of Forms on n-Grids
• Bibliography
• Index

Readership: This monograph is primarily of interest to mathematicians working in the analysis on fractals. It would also be suitable for graduate students as additional reading.