© 2003 by London Mathematical Society
© The London Mathematical Society
Normal and Non-Normal Points of Self-Similar Sets and Divergence Points of Self-Similar Measures
Department of Mathematics, University of St Andrews St Andrews, Fife KY16 9SS, lo{at}st-and.ac.uk
Institut für Mathematik und Informatik, Ernst-Moritz-Arndt-Universität Greifswald D-17487 Greifswald, Germany, winter{at}math-inf.uni-greifswald.de
Received 20 June 2001. Revision received 3 January 2002.
Let K and µ be the self-similar set and the self-similar measure associated with an IFS (iterated function system) with probabilities (Si, pi)i=1,...,N satisfying the open set condition. Let 
={1,...,N}N denote the full shift space and let 
: 
K denote the natural projection. The (symbolic) local dimension of µ at 
is defined by limn (log µK|n/log diam K|n), where 
for = (1, 2,...) . A point for which the limit limn (log µK|n/log diam K|n) does not exist is called a divergence point. In almost all of the literature the limit limn (log µK|n/log diam K|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 (
n)n, let A(n) denote the set of accumulation points of (n)n. For an arbitrary subset I of R, the Hausdorff and packing dimension of the set
 |
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.