Cantor sets with absolutely continuous harmonic measure

Guy David; Cole Jeznach; Antoine Julia

doi:10.5802/jep.245

Cantor sets with absolutely continuous harmonic measure
[Ensembles de Cantor avec une mesure harmonique absolument continue]

Guy David ¹ ; Cole Jeznach ² ; Antoine Julia ¹

¹ Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay 91405 Orsay, France
² School of Mathematics, University of Minnesota Minneapolis, MN, 55455, USA

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1277-1298.

Résumé
Abstract

Nous construisons des ensembles de Cantor $K$ , Ahlfors-réguliers de petite dimension dans le plan, tels que la mesure de Hausdorff sur $K$ est équivalente à la mesure harmonique associée à son complémentaire. En particulier, la fonction de Green dans $ℝ^{2} ∖ K$ satisfait $G^{p} (x) ≃ dist {(x, K)}^{δ}$ lorsque $dist (x, K) \leq 1$ et $p$ est loin de $K$ .

We construct Ahlfors regular Cantor sets $K$ of small dimension in the plane, such that the Hausdorff measure on $K$ is equivalent to the harmonic measure associated to its complement. In particular Green’s function in $ℝ^{2} ∖ K$ satisfies $G^{p} (x) ≃ dist {(x, K)}^{δ}$ whenever $dist (x, K) \leq 1$ and $p$ is far from $K$ .

Reçu le : 2023-04-02
Accepté le : 2023-09-28
Publié le : 2023-10-17

DOI : 10.5802/jep.245

Classification : 31A15, 35J15
Keywords: Harmonic measure, Cantor set, Hausdorff measure
Mot clés : Mesure harmonique, Ensemble de Cantor, mesure de Hausdorff

Affiliations des auteurs :

Guy David ¹ ; Cole Jeznach ² ; Antoine Julia ¹

¹ Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay 91405 Orsay, France
² School of Mathematics, University of Minnesota Minneapolis, MN, 55455, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Guy David and Cole Jeznach and Antoine Julia},
     title = {Cantor sets with absolutely continuous harmonic measure},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1277--1298},
     publisher = {\'Ecole polytechnique},
     volume = {10},
     year = {2023},
     doi = {10.5802/jep.245},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.245/}
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Guy David; Cole Jeznach; Antoine Julia. Cantor sets with absolutely continuous harmonic measure. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1277-1298. doi : 10.5802/jep.245. https://jep.centre-mersenne.org/articles/10.5802/jep.245/

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