Multifractal formalism for typical probability measures on self-similar sets

TL;DR

This paper studies the multifractal properties of typical measures on self-similar sets, showing they follow the multifractal formalism and characterizing their spectra.

Contribution

It computes the multifractal spectrum of typical measures supported on self-similar sets, extending understanding of their multifractal structure.

Findings

Typical measures satisfy the multifractal formalism

The multifractal spectrum of typical measures is explicitly computed

Supports are self-similar sets with well-characterized multifractal properties

Abstract

In this work, we investigate the H\"older spectrum of typical measures (in the Baire category sense) in a general compact set and we compute the multifractal spectrum of a typical measures supported by a self-similar set. Such mesures verify the multifractal formalism.