Skip to main content

2014 | OriginalPaper | Buchkapitel

The Geometry of Fractal Percolation

verfasst von : Michał Rams, Károly Simon

Erschienen in: Geometry and Analysis of Fractals

Verlag: Springer Berlin Heidelberg

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions:
\(\bullet \) the statements work for all directions, not almost all,
\(\bullet \) the statements are true for more general projections, for example radial projections onto a circle,
\(\bullet \) in the case \(\dim _H >1\), each projection has not only positive Lebesgue measure but also has nonempty interior.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literatur
[Ar12]
Zurück zum Zitat Arhosalo, I., Järvenpää, E., Järvenpää, M., Rams, M., Shmerkin, P.: Visible parts of fractal percolation. Proc. Edinburgh Math. Soc. (Series 2), 55(02):311–331 (2012) Arhosalo, I., Järvenpää, E., Järvenpää, M., Rams, M., Shmerkin, P.: Visible parts of fractal percolation. Proc. Edinburgh Math. Soc. (Series 2), 55(02):311–331 (2012)
[CCD88]
Zurück zum Zitat Chayes, J.T., Chayes, L., Durrett, R.: Connectivity properties of mandelbrot’s percolation process. Probability theory and related fields 77(3), 307–324 (1988)CrossRefMATHMathSciNet Chayes, J.T., Chayes, L., Durrett, R.: Connectivity properties of mandelbrot’s percolation process. Probability theory and related fields 77(3), 307–324 (1988)CrossRefMATHMathSciNet
[Ch96]
Zurück zum Zitat Chayes, L.: On the length of the shortest crossing in the super-critical phase of mandelbrot’s percolation process. Stochastic processes and their applications 61(1), 25–43 (1996)CrossRefMATHMathSciNet Chayes, L.: On the length of the shortest crossing in the super-critical phase of mandelbrot’s percolation process. Stochastic processes and their applications 61(1), 25–43 (1996)CrossRefMATHMathSciNet
[DG88]
Zurück zum Zitat Dekking, F.M., Grimmett, G.R.: Superbranching processes and projections of random cantor sets. Probability theory and related fields 78(3), 335–355 (1988)CrossRefMATHMathSciNet Dekking, F.M., Grimmett, G.R.: Superbranching processes and projections of random cantor sets. Probability theory and related fields 78(3), 335–355 (1988)CrossRefMATHMathSciNet
[De09]
Zurück zum Zitat Dekking, M.: Random cantor sets and their projections. Fractal Geom. Stochast. IV, 269–284 (2009) Dekking, M.: Random cantor sets and their projections. Fractal Geom. Stochast. IV, 269–284 (2009)
[DM90]
Zurück zum Zitat Dekking, M., Meester, R.W.J.: On the structure of mandelbrot’s percolation process and other random cantor sets. J. Stat. Phys. 58(5), 1109–1126 (1990)CrossRefMATHMathSciNet Dekking, M., Meester, R.W.J.: On the structure of mandelbrot’s percolation process and other random cantor sets. J. Stat. Phys. 58(5), 1109–1126 (1990)CrossRefMATHMathSciNet
[DS08]
Zurück zum Zitat Dekking, M., Simon, K.: On the size of the algebraic difference of two random cantor sets. Random Struct. Algorithms 32(2), 205–222 (2008)CrossRefMATHMathSciNet Dekking, M., Simon, K.: On the size of the algebraic difference of two random cantor sets. Random Struct. Algorithms 32(2), 205–222 (2008)CrossRefMATHMathSciNet
[FG92]
Zurück zum Zitat Falconer, K.J., Grimmett, G.R.: On the geometry of random cantor sets and fractal percolation. J. Theor. Probab. 5(3), 465–485 (1992) Falconer, K.J., Grimmett, G.R.: On the geometry of random cantor sets and fractal percolation. J. Theor. Probab. 5(3), 465–485 (1992)
[FG94]
Zurück zum Zitat Falconer, K.J., Grimmett, G.R.: Correction: On the geometry of random cantor sets and fractal percolation. J. Theor. Probab. 7(1), 209–210 (1994)CrossRefMATHMathSciNet Falconer, K.J., Grimmett, G.R.: Correction: On the geometry of random cantor sets and fractal percolation. J. Theor. Probab. 7(1), 209–210 (1994)CrossRefMATHMathSciNet
[Ha81]
[KP76]
[Ma74]
Zurück zum Zitat Mandelbrot, B.B.: Intermittent turbulence in self-similar cascades- divergence of high moments and dimension of the carrier. J. Fluid Mech. 62(2), 331–358 (1974)CrossRefMATH Mandelbrot, B.B.: Intermittent turbulence in self-similar cascades- divergence of high moments and dimension of the carrier. J. Fluid Mech. 62(2), 331–358 (1974)CrossRefMATH
[Ma83]
Zurück zum Zitat Mandelbrot, B.B: The fractal geometry of nature/revised and enlarged edition. p. 495. WH Freeman and Co., New York (1983) Mandelbrot, B.B: The fractal geometry of nature/revised and enlarged edition. p. 495. WH Freeman and Co., New York (1983)
[Ma54]
Zurück zum Zitat Marstrand, J.M.: Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. Lond. Math. Soc. 3(1), 257–302 (1954)CrossRefMathSciNet Marstrand, J.M.: Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. Lond. Math. Soc. 3(1), 257–302 (1954)CrossRefMathSciNet
[MW86]
Zurück zum Zitat Mauldin, R.D., Williams, S.C.: Random recursive constructions: asymptotic geometric and topological properties. Trans. Amer. Math. Soc 295(1), 325–346 (1986)CrossRefMATHMathSciNet Mauldin, R.D., Williams, S.C.: Random recursive constructions: asymptotic geometric and topological properties. Trans. Amer. Math. Soc 295(1), 325–346 (1986)CrossRefMATHMathSciNet
[MSS09]
Zurück zum Zitat Mora, P., Simon, K., Solomyak, B.: The lebesgue measure of the algebraic difference of two random cantor sets. Indagationes Mathematicae 20(1), 131–149 (2009)CrossRefMATHMathSciNet Mora, P., Simon, K., Solomyak, B.: The lebesgue measure of the algebraic difference of two random cantor sets. Indagationes Mathematicae 20(1), 131–149 (2009)CrossRefMATHMathSciNet
[RS00]
Zurück zum Zitat Rams, M., Simon, K.: The dimension of projections of fractal percolations. J. Stat. Phys 154(3), 633–655 (2014) Rams, M., Simon, K.: The dimension of projections of fractal percolations. J. Stat. Phys 154(3), 633–655 (2014)
Metadaten
Titel
The Geometry of Fractal Percolation
verfasst von
Michał Rams
Károly Simon
Copyright-Jahr
2014
Verlag
Springer Berlin Heidelberg
DOI
https://doi.org/10.1007/978-3-662-43920-3_11