Symbolic and geometric local dimensions of self-affine multifractal sierpinski sponges in R-d

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Abstract

In this paper we study the multifractal structure of a certain class of self-affine measures known as self-affine multifractal Sierpinski sponges. Multifractal analysis studies the local scaling behaviour of measures. In particular, multifractal analysis studies the so-called local dimension and the multifractal spectrum of measures. The multifractal structure of self-similar measures satisfying the Open Set Condition is by now well understood. However, the multifractal structure of self-affine multifractal Sierpinski sponges is significantly less well understood. The local dimensions and the multifractal spectrum of self-affine multifractal Sierpinski sponges are only known provided a very restrictive separation condition, known as the Very Strong Separation Condition (VSSC), is satisfied. In this paper we investigate the multifractal structure of general self-affine multifractal Sierpinski sponges without assuming any additional conditions (and, in particular, without assuming the VSSC).

Original languageEnglish
Pages (from-to)37-51
Number of pages15
JournalStochastics and Dynamics
Volume7
Issue number1
DOIs
Publication statusPublished - Mar 2007

Keywords

  • multifractals
  • self-affine measures
  • local dimension
  • HAUSDORFF DIMENSION
  • SETS
  • CARPETS
  • FRACTALS

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