The Mandelbrot–Kahane problem of Benoît Mandelbrot model of turbulence

Xinxin Chen; Yong Han; Yanqi Qiu; Zipeng Wang

doi:10.5802/crmath.697

Article de recherche - Analyse fonctionnelle, Probabilités

The Mandelbrot–Kahane problem of Benoît Mandelbrot model of turbulence
[Le problème de Mandelbrot–Kahane du modèle de turbulence de Benoît Mandelbrot]

Xinxin Chen ¹ ; Yong Han ² ; Yanqi Qiu ³ ; Zipeng Wang ⁴

¹ School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
² School of Mathematical Sciences, Shenzhen University, Shenzhen 518060, Guangdong, China
³ School of Fundamental Physics and Mathematical Sciences, HIAS, University of Chinese Academy of Sciences, Hangzhou 310024, China
⁴ College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 35-41.

Résumé
Abstract

Le but de cet article est d’annoncer notre résolution du problème de Mandelbrot–Kahane (Mandelbrot, 1976 et Kahane, 1993) sur la détermination de la dimension de Fourier de la mesure en cascade canonique de Mandelbrot (MCCM). Plus précisément, nous obtenons la formule exacte pour la dimension de Fourier de la MCCM avec des poids aléatoires $W$ satisfaisant la condition $𝔼 [W^{t}] < \infty$ pour tout $t \geq 1$ . De plus, nous démontrons que la MCCM est une mesure de Rajchman avec une décroissance polynomiale de Fourier lorsque le poids aléatoire satisfait $𝔼 [W^{1 + δ}] < \infty$ pour un certain $δ > 0$ . Dans cette annonce, nous soulignons brièvement les deux applications suivantes : (1) dans le cas frontière de Biggins–Kyprianou, la dimension de Fourier de la MCCM présente une transition de phase du second ordre à la température inverse $β = \frac{1}{2}$ , et (2) la régularité de Frostman supérieure et l’estimation de restriction de Fourier pour la MCCM.

The purpose of this paper is to announce our solution to the Mandelbrot–Kahane problem (Mandelbrot, 1976 and Kahane, 1993) of determining the Fourier dimension of the Mandelbrot canonical cascade measure (MCCM). Specifically, we obtain the exact formula for the Fourier dimension of the MCCM with random weights $W$ satisfying the condition $𝔼 [W^{t}] < \infty$ for all $t \geq 1$ . In addition, we show that the MCCM is Rajchman with polynomial Fourier decay whenever the random weight satisfies $𝔼 [W^{1 + δ}] < \infty$ for some $δ > 0$ . In this announcement, we briefly highlight the following two applications: (1) in the Biggins–Kyprianou’s boundary case, the Fourier dimension of the MCCM exhibits a second order phase transition at the inverse temperature $β = \frac{1}{2}$ , and (2) the upper Frostman regularity and Fourier restriction estimate of the MCCM.

Reçu le : 2024-09-13
Révisé le : 2024-10-13
Accepté le : 2024-10-22
Publié le : 2025-03-06

DOI : 10.5802/crmath.697

Classification : 60G57, 42A61, 46B09, 60J80, 60G46
Keywords: Fourier dimension, Mandelbrot cascades, Rajchman measure, Salem measure
Mots-clés : Dimension de Fourier, cascades de Mandelbrot, mesure de Rajchman, mesure de Salem

Affiliations des auteurs :

Xinxin Chen ¹ ; Yong Han ² ; Yanqi Qiu ³ ; Zipeng Wang ⁴

¹ School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
² School of Mathematical Sciences, Shenzhen University, Shenzhen 518060, Guangdong, China
³ School of Fundamental Physics and Mathematical Sciences, HIAS, University of Chinese Academy of Sciences, Hangzhou 310024, China
⁴ College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Xinxin Chen and Yong Han and Yanqi Qiu and Zipeng Wang},
     title = {The {Mandelbrot{\textendash}Kahane} problem of {Beno{\^\i}t} {Mandelbrot} model of turbulence},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {35--41},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.697},
     language = {en},
}

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Xinxin Chen; Yong Han; Yanqi Qiu; Zipeng Wang. The Mandelbrot–Kahane problem of Benoît Mandelbrot model of turbulence. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 35-41. doi : 10.5802/crmath.697. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.697/

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