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2 results

  • DIMENSIONS OF INTERSECTIONS OF THE SIERPINSKI CARPET WITH LINES OF RATIONAL SLOPES

  • Qing-Hui Liu, Li-Feng Xi, Yan-Fen Zhao
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 50 / Issue 2 / June 2007
    Published online by Cambridge University Press:
    17 May 2007, pp. 411-427
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  • This paper computes the Box and Hausdorff dimensions of the intersections of the Sierpinski carpet with planar lines of rational slopes.

  • Brownian Motion and Harmonic Analysis on Sierpinski Carpets

  • Martin T. Barlow, Richard F. Bass
  • Journal:
    Canadian Journal of Mathematics / Volume 51 / Issue 4 / 01 August 1999
    Published online by Cambridge University Press:
    20 November 2018, pp. 673-744
    Print publication:
    01 August 1999
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  • We consider a class of fractal subsets of ${{\mathbb{R}}^{d}}$ formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion $X$ and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.