The graph, range and level set singularity spectra of $b$-adic independent cascade function

TL;DR

This paper computes the multifractal singularity spectra of the graph, range, and level sets of the $b$-adic independent cascade function, revealing detailed fractal properties of these self-similar functions.

Contribution

It provides the first detailed analysis of the singularity spectra for the graph, range, and level sets of $b$-adic independent cascade functions, a class of multifractal functions.

Findings

Explicit formulas for the singularity spectra of the graph, range, and level sets.

Demonstrates the multifractal nature of $b$-adic independent cascade functions.

Enhances understanding of the geometric complexity of multifractal functions.

Abstract

With the "iso-H\"older" sets of a function we naturally associate subsets of the graph, range and level set of the function. We compute the associated singularity spectra for a class of statistically self-similar multifractal functions, namely the $b$-adic independent cascade function.