Multifractal analysis of complex random cascades

TL;DR

This paper develops a multifractal analysis method for complex-valued self-similar functions, revealing new phenomena such as functions obeying the multifractal formalism with a singularity spectrum spanning all positive real numbers.

Contribution

It introduces a novel multifractal formalism for complex functions and demonstrates cases with full-spectrum support, advancing understanding of multifractality in continuous functions.

Findings

Functions obey the multifractal formalism.

Support of the singularity spectrum is the entire [0, ∞] interval.

New phenomena in multifractal analysis of continuous functions.

Abstract

We achieve the multifractal analysis of a class of complex valued statistically self-similar continuous functions. For we use multifractal formalisms associated with pointwise oscillation exponents of all orders. Our study exhibits new phenomena in multifractal analysis of continuous functions. In particular, we find examples of statistically self-similar such functions obeying the multifractal formalism and for which the support of the singularity spectrum is the whole interval $[0,\infty]$.