Geometriae Dedicata

, Volume 159, Issue 1, pp 307–325 | Cite as

Minkowski content and local Minkowski content for a class of self-conformal sets

Original Paper


We investigate (local) Minkowski measurability of \({\mathcal {C}^{1+\alpha}}\) images of self-similar sets. We show that (local) Minkowski measurability of a self-similar set K implies (local) Minkowski measurability of its image F and provide an explicit formula for the (local) Minkowski content of F in this case. A counterexample is presented which shows that the converse is not necessarily true. That is, F can be Minkowski measurable although K is not. However, we obtain that an average version of the (local) Minkowski content of both K and F always exists and also provide an explicit formula for the relation between the (local) average Minkowski contents of K and F.


Minkowski content Conformal iterated function system Self-conformal set Fractal curvature measures 

Mathematics Subject Classification (2010)

28A80 28A75 


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  1. 1.
    Benedetti R., Petronio C.: Lectures on Hyperbolic Geometry. Universitext. Springer, Berlin (1992)CrossRefGoogle Scholar
  2. 2.
    Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. 2nd revised ed. Lecture Notes in Mathematics 470. Springer, Berlin (2008)Google Scholar
  3. 3.
    Connes A.: Noncommutative Geometry. Academic Press, San Diego, CA (1994)MATHGoogle Scholar
  4. 4.
    Deniz, A., Kocak, S., Özdemir, Y., Ratiu, A., Üreyen, A.: On the Minkowski measurability of self-similar fractals in \({\mathbb{R}^d}\) . Preprint: arXiv:1006.5883v1 (2010)Google Scholar
  5. 5.
    Falconer K.: On the Minkowski measurability of fractals. Proc. Am. Math. Soc. 123(4), 1115–1124 (1995)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Falconer K.: Fractal Geometry, Mathematical Foundations and Applications. Wiley, Chichester (2003)MATHCrossRefGoogle Scholar
  7. 7.
    Falconer K., Samuel T.: Dixmier traces and coarse multifractal analysis. Ergodic Theory Dyn. Syst. 31(2), 369–381 (2011)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Gatzouras D.: Lacunarity of self-similar and stochastically self-similar sets. Trans. Am. Math. Soc. 352(5), 1953–1983 (2000)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Guido D., Isola T.: Dimensions and singular traces for spectral triples, with applications to fractals. J. Funct. Anal. 203(2), 362–400 (2003)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Kesseböhmer, M., Kombrink, S.: Fractal curvature measures and Minkowski content for self-conformal subsets of the real line. Preprint: arXiv:1012.5399v2 (2010)Google Scholar
  11. 11.
    Lapidus M.: Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture. Trans. Am. Math. Soc. 325(2), 465–529 (1991)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Lapidus M., van Frankenhuijsen M.: Fractal Geometry, Complex Dimensions and Zeta Functions Geometry and Spectra of Fractal Strings. Springer, New York (2006)MATHGoogle Scholar
  13. 13.
    Lapidus, M., Pearse, E., Winter, S.: Minkowski measurability results for self-similar tilings and fractals with monophase generators. Preprint: arXiv:1104.1641v1 (2011)Google Scholar
  14. 14.
    Lapidus M., Pomerance C.: The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums. Proc. Lond. Math. Soc. (3) 66(1), 41–69 (1993)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Levitin M., Vassiliev D.: Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals. Proc. Lond. Math. Soc. III. Ser. 72(1), 188–214 (1996)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Mandelbrot B.: The Fractal Geometry of Nature. W. H. Freeman and Co., San Francisco (1982)MATHGoogle Scholar
  17. 17.
    Mandelbrot, B.: Measures of fractal lacunarity: Minkowski content and alternatives. In: Fractal Geometry and Stochastics. Birkhäuser Verlag, Basel (1995)Google Scholar
  18. 18.
    Mauldin R., Urbański M.: Dimensions and measures in infinite iterated function systems. Proc. Lond. Math. Soc. III. Ser. 73(1), 105–154 (1996)MATHCrossRefGoogle Scholar
  19. 19.
    Samuel, T.: A commutative noncommutative fractal geometry. Ph.D. thesis, University of St Andrews (2010)Google Scholar
  20. 20.
    Schief A.: Separation properties for self-similar sets. Proc. Am. Math. Soc. 122(1), 111–115 (1994)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Winter, S.: Curvature measures and fractals. Diss. Math. 453 (2008)Google Scholar
  22. 22.
    Winter, S., Zähle, M.: Fractal curvature measures of self-similar sets. Preprint: arXiv:1007.0696v2 (2010)Google Scholar
  23. 23.
    Zähle M.: Lipschitz-Killing curvatures of self-similar random fractals. Trans. Am. Math. Soc. 363(5), 2663–2684 (2011)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Universität Siegen, FB 6—MathematikSiegenGermany
  2. 2.Universität Bremen, FB 3—MathematikBremenGermany

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