On multifractality and time subordination for continuous functions

TL;DR

This paper demonstrates that homogeneously multifractal functions can be decomposed into a monofractal function composed with a time subordinator, offering new insights into their multifractal structure.

Contribution

It establishes a unique decomposition of homogeneously multifractal functions into monofractal functions and time subordinators, deepening understanding of multifractal behaviors.

Findings

Homogeneously multifractal functions can be expressed as compositions of monofractal functions and time subordinators.

The monofractality exponent of the associated function is uniquely determined.

Analysis of classical examples illustrates the decomposition and its implications.

Abstract

We show that if $Z$ is "homogeneously multifractal" (in a sense we precisely define), then $Z$ is the composition of a monofractal function $g$ with a time subordinator $f$ (i.e. $f$ is the integral of a positive Borel measure supported by $\zu$). When the initial function $Z$ is given, the monofractality exponent of the associated function $g$ is uniquely determined. We study in details a classical example of multifractal functions $Z$, for which we exhibit the associated functions $g$ and $f$. This provides new insights into the understanding of multifractal behaviors of functions.