[Paper Review] A mixed multifractal formalism for finitely many non Gibbs Frostman-like measures

Mohamed Menceur, Anouar Ben Mabrouk|arXiv (Cornell University)|Apr 21, 2018

Mathematical Dynamics and Fractals18 references3 citations

TL;DR

This paper introduces a generalized mixed multifractal formalism for finitely many non-Gibbs, Frostman-like measures by replacing the standard Gibbs measure assumption with a control function, allowing the multifractal formalism to hold under broader conditions. The key contribution is a new large deviation theorem that enables the derivation of dimension spectra even when traditional Gibbs-like measures do not apply.

ABSTRACT

The multifractal formalism for measures hold whenever the existence of corresponding Gibbs-like measures supported on the singularities sets holds. In the present work we tried to relax such a hypothesis and introduce a more general framework of mixed (and thus single) multifractal analysis where the measures constructed on the singularities sets are not Gibbs but controlled by an extra-function allowing the multifractal formalism to hold. We fall on the classical case by a particular choice of such a function.

Motivation & Objective

• To relax the Gibbs measure assumption in multifractal formalism, which traditionally requires the existence of Gibbs-like measures on level sets.
• To develop a mixed multifractal framework where measures on singularity sets are not Gibbs but controlled by an auxiliary function.
• To establish a multifractal formalism that holds even when the standard Gibbs condition fails, by introducing a φ-controlled generalization of Hausdorff and packing measures.
• To generalize existing multifractal dimension definitions (Hausdorff, packing, logarithmic index) using vector-valued measures and a parameterized pre-measure construction.
• To prove a new mixed large deviation theorem that supports the validity of the formalism in the absence of Gibbs measures.

Proposed method

• Introduces φ-mixed multifractal pre-measures using a function φ to control the scaling behavior of measures on singularity sets, generalizing standard Hausdorff and packing measures.
• Defines generalized dimensions via the critical exponent t where the φ-mixed pre-measure transitions from infinity to zero, extending the classical multifractal formalism.
• Applies a vector-valued large deviation theorem (Theorem 5.2) to analyze the asymptotic behavior of empirical measures associated with level sets.
• Uses the Legendre transform and convex analysis to relate the free energy function C(t) to the scaling behavior of measure moments.
• Employs the Borel-Cantelli lemma and exponential estimates to prove almost sure convergence of normalized empirical measures to the Legendre transform of the free energy.
• Relies on the Besicovitch covering theorem and measure-theoretic covering arguments to ensure the validity of the dimension estimates.

Experimental results

Research questions

• RQ1 Can the multifractal formalism be extended to measures that are not Gibbs-like, by replacing the Gibbs assumption with a more general control function?
• RQ2 How can generalized Hausdorff and packing measures be constructed to reflect the local scaling behavior of non-Gibbs measures?
• RQ3 What conditions on the control function φ ensure the validity of the multifractal formalism in mixed settings?
• RQ4 Can a large deviation principle be established for vector-valued empirical measures in the absence of Gibbs properties?
• RQ5 What is the relationship between the generalized dimension spectrum and the Legendre transform of the free energy function in this extended framework?

Key findings

• The multifractal formalism holds for finitely many non-Gibbs, Frostman-like measures when the singularity set measures are controlled by a function φ, generalizing the classical case.
• The φ-mixed multifractal pre-measures $ar{rak{H}}_{ u}^{q,t}$ and $ar{rak{P}}_{ u}^{q,t}$ are defined via infima and suprema over coverings and packings, respectively, with scaling controlled by φ.
• The generalized dimension functions $ ext{dim}_{ u}^{q}(E)$, $ ext{Dim}_{ u}^{q}(E)$, and $ riangle_{ u}^{q}(E)$ are shown to satisfy the standard cut-off property: the measure is infinite below the critical dimension and zero above.
• A new large deviation theorem (Theorem 5.2) is proven, showing that the normalized empirical measures converge almost surely to the Legendre transform of the free energy function C(t).
• The result $ abla_{-}C(0) riangleq ext{ess} ext{inf}igl(rac{W_n}{a_n}igr) ext{ a.s.}$ and $ abla_{+}C(0) riangleq ext{ess} ext{sup}igl(rac{W_n}{a_n}igr) ext{ a.s.}$ is established under summability conditions on $e^{- u a_n}$, ensuring concentration of measure.
• The classical multifractal formalism is recovered as a special case when the control function φ is chosen to be the identity, confirming consistency with existing theory.