Mixed divergence points for self-similar measures

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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


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.,


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


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


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