Skip to main content
SpringerLink
Log in

Asymptotic porosity of planar harmonic measure

  • Published:
Arkiv för Matematik

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

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Graczyk, J. and Świa̧tek, G., Harmonic measure and expansion on the boundary of the connectedness locus, Invent. Math. 142 (2000), 605–629.

    Article  MathSciNet  MATH  Google Scholar 

  2. Koskela, P. and Rohde, S., Hausdorff dimension and mean porosity, Math. Ann. 309 (1997), 593–609.

    Article  MathSciNet  MATH  Google Scholar 

  3. Makarov, N. G., Distortion of boundary sets under conformal mappings, Proc. Lond. Math. Soc. 51 (1985), 369–384.

    Article  MATH  Google Scholar 

  4. Mattila, P., Distribution of sets and measures along planes, J. Lond. Math. Soc. 38 (1988), 125–132.

    Article  MathSciNet  MATH  Google Scholar 

  5. Mattila, P., Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, Cambridge, 1995.

    Book  MATH  Google Scholar 

  6. McMullen, C. T., Renormalization and 3-Manifolds which Fiber over the Circle, Annals of Mathematics Studies 142, Princeton University Press, Princeton, NJ, 1996.

    MATH  Google Scholar 

  7. McMullen, C. T., The Mandelbrot set is universal, in The Mandelbrot Set, Theme and Variations, Lond. Mat. Soc. Lecture Series 274, Cambridge University Press, Cambridge, 2000.

    Google Scholar 

  8. Shishikura, M., The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Ann. of Math. 147 (1998), 225–267.

    Article  MathSciNet  MATH  Google Scholar 

  9. Smirnov, S., Symbolic dynamics and Collet–Eckmann conditions, Int. Math. Res. Not. 7 (2000), 333–351.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Paris XI, FR-91405, Orsay Cedex, France

    Jacek Graczyk

  2. Department of Mathematics and Information Sciences, Warsaw University of Technology, PL-00-661, Warszawa, Poland

    Grzegorz Świa̧tek

Authors
  1. Jacek Graczyk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Grzegorz Świa̧tek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jacek Graczyk.

Additional information

Partial support from the Research Training Network CODY is acknowledged.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Graczyk, J., Świa̧tek, G. Asymptotic porosity of planar harmonic measure. Ark Mat 51, 53–69 (2013). https://doi.org/10.1007/s11512-011-0154-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11512-011-0154-4

Keywords

Search

Navigation