Iterated function systems with a weak separation condition
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AbstractNonoverlapping contractive self-similar iterated function systems (IFS) have been studied in great detail via the open set condition. On the other hand much less is known about IFS with overlaps. To deal with such systems, a weak separation condition (WSC) has been introduced recently [LN1]; it is weaker than the open set condition and it includes many important overlapping cases. This paper has two purposes. First, we consider the class of self-similar measures generated by such IFS; we give a necessary and sufficient condition for the self-similar measures to be absolutely continuous with respect to Hausdorff measures. This extends a result in [LNR]. As most of the known examples of the WSC involve algebraic integers (e.g., the golden ratio, integral dilation matrices) and the contraction ratios are equal, our second goal is to give new examples of self-similar IFS with the WSC and with more arbitrary contraction ratios.
All Author(s) ListLau KS, Wang XY
Journal nameStudia Mathematica
Volume Number161
Issue Number3
Pages249 - 268
LanguagesEnglish-United Kingdom
Keywordsabsolute continuity; iterated function system; self-similar measure; weak separation condition
Web of Science Subject CategoriesMathematics; MATHEMATICS

Last updated on 2021-23-02 at 00:23