## Arkiv för Matematik

### Asymptotic porosity of planar harmonic measure

#### Abstract

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 λn for n from a set with positive density amongst natural numbers.

#### Note

Partial support from the Research Training Network CODY is acknowledged.

#### Article information

Source
Ark. Mat., Volume 51, Number 1 (2013), 53-69.

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485907197

Digital Object Identifier
doi:10.1007/s11512-011-0154-4

Mathematical Reviews number (MathSciNet)
MR3029336

Rights

#### Citation

Graczyk, Jacek; Świa̧tek, Grzegorz. Asymptotic porosity of planar harmonic measure. Ark. Mat. 51 (2013), no. 1, 53--69. doi:10.1007/s11512-011-0154-4. https://projecteuclid.org/euclid.afm/1485907197

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