Normal and non-normal points of self-similar sets and divergence points of self-similar measures

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88 Citations (Scopus)

Abstract

Let K and mu be the self-similar set and the self-similar measure associated with an IFS (iterated function system) with probabilities (S-i,p(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 at omega is an element of Sigma is defined by lim(n) (log muK(omega\n)/log diam K-omega\n), where K-omega\n = S-omega1 o...o S-omegan (K) for omega = (omega1,omega2,...) is an element of Sigma. A point omega for which the limit lim(n) (log muK(omega\n)/log diam K-omega\n) does not exist is called a divergence point. In almost all of the literature the limit lim(n) (log muK(omega\n)/log diam K-omega\n) is assumed to exist, and almost nothing is known about the set of divergence points. In the paper a detailed analysis is performed of the set of divergence points and it is shown that it has 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 I of R, the Hausdorff and packing dimension of the set

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and related sets is computed. An interesting and surprising corollary to this result is that the set of divergence points is extremely 'visible'; it can be partitioned into an uncountable family of pairwise disjoint sets each with full dimension. In order to prove the above statements the theory of normal and non-normal points of a self-similar set is formulated and developed in detail. This theory extends the notion of normal and non-normal numbers to the setting of self-similar sets and has numerous applications to the study of the local properties of self-similar measures including a detailed study of the set of divergence points.

Original languageEnglish
Pages (from-to)103-122
Number of pages20
JournalJournal of the London Mathematical Society
Volume67
Issue number1
DOIs
Publication statusPublished - Feb 2003

Keywords

  • HAUSDORFF DIMENSION
  • FRACTALS

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