Asymptotic porosity of planar harmonic measure

Jacek Graczyk Department of Mathematics, University of Paris XI Grzegorz Świa̧tek Department of Mathematics and Information Sciences, Warsaw University of Technology

Classical Analysis and ODEs mathscidoc:1701.05004

Arkiv for Matematik, 51, (1), 53-69, 2010.9
We study the distribution of harmonic measure on connected Julia sets of unicritical polynomials. Harmonic measure on a full compact set in ℂ is always concentrated on a set which is porous for a positive density of scales. We prove that there is a topologically generic set $\mathcal{A}$ in the boundary of the Mandelbrot set such that for every $c\in \mathcal{A}$ ,$β$>0, and$λ$∈(0,1), the corresponding Julia set is a full compact set with harmonic measure concentrated on a set which is not$β$-porous in scale$λ$^{$n$}for$n$from a set with positive density amongst natural numbers.
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  title={Asymptotic porosity of planar harmonic measure},
  author={Jacek Graczyk, and Grzegorz Świa̧tek},
  booktitle={Arkiv for Matematik},
Jacek Graczyk, and Grzegorz Świa̧tek. Asymptotic porosity of planar harmonic measure. 2010. Vol. 51. In Arkiv for Matematik. pp.53-69.
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