Normal and Non-Normal Points of Self-Similar Sets and Divergence Points of Self-Similar Measures |
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Authors: | Olsen L; Winter S |
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Institution: | 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 |
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Abstract: | Let K and µ be the self-similar set and the self-similarmeasure associated with an IFS (iterated function system) withprobabilities (Si, pi)i=1,...,N satisfying the open set condition.Let ={1,...,N}N denote the full shift space and let : K denotethe natural projection. The (symbolic) local dimension of µat 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) doesnot exist is called a divergence point. In almost all of theliterature the limit limn (log µK|n/log diam K|n) is assumedto exist, and almost nothing is known about the set of divergencepoints. In the paper a detailed analysis is performed of theset of divergence points and it is shown that it has a surprisinglyrich structure. For a sequence (n)n, let A(n) denote the setof accumulation points of (n)n. For an arbitrary subset I ofR, the Hausdorff and packing dimension of the set
and related sets is computed. An interesting and surprisingcorollary to this result is that the set of divergence pointsis extremely visible; it can be partitioned intoan uncountable family of pairwise disjoint sets each with fulldimension. In order to prove the above statements the theory of normaland non-normal points of a self-similar set is formulated anddeveloped in detail. This theory extends the notion of normaland non-normal numbers to the setting of self-similar sets andhas numerous applications to the study of the local propertiesof self-similar measures including a detailed study of the setof divergence points. |
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