Composition of Fractals

TL;DR

This paper investigates how combining multiple fractals affects their overall dimension, providing analytical and numerical methods to understand the composite dimension, especially when involving multifractals, with applications in physics.

Contribution

It introduces a framework for analyzing the composition of fractals, including multifractals, and explores the relation between component and composite dimensions.

Findings

Composite dimension for standard IFS can be expressed analytically.

Composite dimension involving multifractals requires numerical solutions.

Application demonstrated in a physics context using incomplete statistics.

Abstract

This work is an analytical and numerical study of the composition of several fractals into one and of the relation between the composite dimension and the dimensions of the component fractals. In the case of composition of standard IFS with segments of equal size, the composite dimension can be expressed as a function of the component dimensions. But in the case of the compositions including component multifractals, the composite dimension cannot be expressed as explicit function of component dimensions and can only be solved numerically. An application of fractal composition to a physics problem within the incomplete statistics is discussed.