Mixed divergence points for self-similar measures

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18 Citations (Scopus)

Abstract

For j = 1,..., k, let K and mu(j) be the self-similar set and the self-similar measure associated with an ITS with probabilities (S-i, p(j,i)) (i= 1,...,N) satisfying the Open Set Condition. Let Sigma = {1,..., N }(N) denote the full shift space and let pi : Sigma --> K denote the natural projection. The (symbolic) local dimension of mu(j) at omega is an element of Sigma is defined by lim(n) (log mu(j)K(omega\n) / log diam K-omega\n), where K-omega\n = S-omega1 circle (...) circle S-omegan(K) for omega = omega(1)omega(2)... is an element of Sigma. A point omega for which the limit lim(n) (log mu(j)K(omega\n)/ log diam K-omega\n) does not exist is called a divergence point for muj. Previously only divergence points of a single measure have been investigated. In this paper we perform a detailed analysis of sets of points that are divergence points for all the measures mu(1),..., mu(k) simultaneously, and show that these points have a surprisingly rich structure. For a sequence (x(n))(n), let A (x(n)) denote the set of accumulation points of (x(n))(n). For an arbitrary subset C of R-k, we compute the Hausdorff and packing dimensions of the set

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and related sets. An interesting and surprising corollary to our result is that the set of simultaneous divergence points is extremely "visible", namely, (typically) it has full Hausdorff dimension, i.e.,

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Original languageEnglish
Pages (from-to)1343-1372
Number of pages30
JournalIndiana University Mathematics Journal
Volume52
Publication statusPublished - 2003

Keywords

  • fractals
  • multifractals
  • mixed multifractal spectrum
  • Hausdorff measure
  • packing measure
  • divergence points
  • local dimension
  • self-similar measure
  • HAUSDORFF DIMENSION
  • SETS
  • FRACTALS

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