Analysis on hybrid fractals

TL;DR

This paper introduces hybrid fractals formed by gluing fractal pieces, studies their energy forms and Laplacians, and analyzes spectral properties both theoretically and numerically, revealing energies that do not fully capture the fractal structure.

Contribution

It constructs explicit energy forms on hybrid fractals, especially on the 3-level Sierpinski gasket, and investigates their spectral properties within a general framework of finitely ramified structures.

Findings

Energy forms that do not fully capture the fractal structure are constructed.

Spectral analysis reveals asymptotic behavior of Laplacians on hybrid fractals.

Numerical data supports theoretical spectral analysis.

Abstract

We introduce hybrid fractals as a class of fractals constructed by gluing several fractal pieces in a specific manner and study energy forms and Laplacians on them. We consider in particular a hybrid based on the $3$-level Sierpinski gasket, for which we construct explicitly an energy form with the property that it does not "capture" the $3$-level Sierpinski gasket structure. This characteristic type of energy forms that "miss" parts of the structure of the underlying space are investigated in the more general framework of finitely ramified cell structures. The spectrum of the associated Laplacian and its asymptotic behavior in two different hybrids is analyzed theoretically and numerically. A website with further numerical data analysis is available at http://www.math.cornell.edu/~harry970804/.