Journal of Theoretical Probability

, Volume 32, Issue 2, pp 608–632 | Cite as

Porosities of Mandelbrot Percolation

  • Artemi BerlinkovEmail author
  • Esa Järvenpää


We study porosities in the Mandelbrot percolation process using a notion of porosity that is based on the construction geometry. We show that, almost surely at almost all points with respect to the natural measure, the construction-based mean porosities of the set and the natural measure exist and are equal to each other for all parameter values outside of a countable exceptional set. As a corollary, we obtain that, almost surely at almost all points, the regular lower porosities of the set and the natural measure are equal to zero, whereas the regular upper porosities reach their maximum values.


Random sets Porosity Mean porosity 

Mathematics Subject Classification (2010)

28A80 37A50 60D05 60J80 



We thank Maarit Järvenpää for interesting discussions and many useful comments. We are also thankful to the referee for the comments and suggestions on improvement in this paper.


  1. 1.
    Athreya, K., Ney, P.: Branching Processes. Springer-Verlag, Berlin (1972)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bernstein, S.N.: On the law of large numbers. Soob. Khar’k Mat. Ob-va 2(16), 82–87 (1918). (in Russian)Google Scholar
  3. 3.
    Beliaev, D., Smirnov, S.: On dimension of porous measures. Math. Ann. 323, 123–141 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beliaev, D., Järvenpää, E., Järvenpää, M., Käenmäki, A., Rajala, T., Smirnov, S., Suomala, V.: Packing dimension of mean porous measures. J. Lond. Math. Soc. 80, 514–530 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berlinkov, A.: On random fractals with infinite branching: definition, measurability, dimensions. Ann. Inst. H. Poincarè Probab. Stat. 49, 1080–1089 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berlinkov, A.: Exact packing dimension in random recursive constructions. Probab. Theor. Relat. Fields 126, 477–496 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, C., Ojala, T., Rossi, E., Suomala, V.: Fractal percolation, porosity and dimension. J. Theor. Probab. 30, 1471–1498 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Denjoy, A.: Sur une propriété des séries trigonométriques, Verlag v.d.G.V. der Wis-en Natuur. Afd. (1920)Google Scholar
  9. 9.
    Dolženko, E.P.: Boundary properties of arbitrary functions. Izv. Akad. Nauk SSSR Ser. Mat. 31, 3–14 (1967). (in Russian)MathSciNetGoogle Scholar
  10. 10.
    Dubuc, S., Seneta, E.: The local limit theorem for the Galton–Watson process. Ann. Probab. 4, 490–496 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Elliott, E.O.: Measures on product spaces. Trans. Am. Math. Soc. 128, 379–388 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Eckmann, J.-P., Järvenpää, E., Järvenpää, M.: Porosities and dimensions of measures. Nonlinearity 13, 1–18 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gnedenko, B.V.: The Theory of Probability, (translated by B. D. Sechler), Chelsea Publishing Company, New York (1969)Google Scholar
  14. 14.
    Graf, S., Mauldin, R.D., Williams, S.: The exact Hausdorff dimension in random recursive constructions. Mem. Am. Math. Soc. 381, 1–121 (1988)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hu, T.-C., Rosalsky, A., Volodin, A.: On convergence properties of sums of dependent random variables under second moment and covariance restrictions. Stat. Probab. Lett. 78, 1999–2005 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Järvenpää, E., Järvenpää, M.: Porous measures on \({\mathbb{R}}^n\): local structure and dimensional properties. Proc. Am. Math. Soc. 130, 419–426 (2001)CrossRefzbMATHGoogle Scholar
  17. 17.
    Järvenpää, E., Järvenpää, M.: Average homogeneity and dimensions of measures. Math. Ann. 331, 557–576 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Järvenpää, E., Järvenpää, M., Käenmäki, A., Rajala, T., Rogovin, S., Suomala, V.: Packing dimension and Ahlfors regularity of porous sets in metric spaces. Math. Z. 266, 83–105 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Järvenpää, E., Järvenpää, M., Käenmäki, A., Suomala, V.: Asymptotically sharp dimension estimates for \(k\)-porous sets. Math. Scand. 97, 309–318 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Järvenpää, E., Järvenpää, M., Mauldin, R.D.: Deterministic and random aspects of porosities. Discrete Contin. Dyn. Syst. 8, 121–136 (2002)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Käenmäki, A., Rajala, T., Suomala, V.: Local homogeneity and dimensions of measures in doubling metric spaces. An. Sc. Norm. Super. Pisa Cl. Sci. (5) XVI, 1315–1351 (2016)zbMATHGoogle Scholar
  22. 22.
    Käenmäki, A., Suomala, V.: Conical upper density theorems and porosity of measures. Adv. Math. 217, 952–966 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Käenmäki, A., Suomala, V.: Nonsymmetric conical upper density and \(k\)-porosity. Trans. Am. Math. Soc. 363, 1183–1195 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kahane, J.-P., Peyrière, J.: Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22, 131–145 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Koskela, P., Rohde, S.: Hausdorff dimension and mean porosity. Math. Ann. 309, 593–609 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lyons, R.: Strong law of large numbers for weakly correlated random variables. Mich. Math. J. 35, 353–359 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mandelbrot, B.: Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331–358 (1974)CrossRefzbMATHGoogle Scholar
  28. 28.
    Mattila, P.: Distribution of sets and measures along planes. J. Lond. Math. Soc. 38, 125–132 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  30. 30.
    Mauldin, R.D., Williams, S.C.: Random recursive constructions: asymptotic geometric and topological properties. Trans. Am. Math. Soc. 295, 325–345 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Mera, M.E., Morán, M.: Attainable values for upper porosities of measures. Real Anal. Exch. 26, 101–115 (2000/01)Google Scholar
  32. 32.
    Mera, M.E., Morán, M., Preiss, D., Zajíček, L.: Porosity, \(\sigma \)-porosity and measures. Nonlinearity 16, 247–255 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Salli, A.: On the Minkowski dimension of strongly porous fractal sets in \({\mathbb{R}}^n\). Proc. Lond. Math. Soc. 62, 353–372 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Shmerkin, P.: Porosity, dimension, and local entropies: a survey. Rev. Un. Mat. Argentina 52, 81–103 (2011)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Shmerkin, P.: The dimension of weakly mean porous measures: a probabilistic approach. Int. Math. Res. Not. IMRN 9, 2010–2033 (2012)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Urbański, M.: Porosity in conformal iterated function systems. J. Number Theory 88, 283–312 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Zajíček, L.: Porosity and \(\sigma \)-porosity. Real Anal. Exch. 13, 314–350 (1987–1988)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat GanIsrael
  2. 2.University ITMOSt. PetersburgRussian Federation
  3. 3.Department of Mathematical SciencesUniversity of OuluOuluFinland

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