Iterated function systems with a weak separation condition

Ka-Sing Lau ¹   ; Xiang-Yang Wang ²

¹ Department of Mathematics The Chinese University of Hong Kong, H.K.
² Department of Mathematics The Chinese University of Hong Kong, H.K. and School of Mathematics and Computational Science Zhong-Shan University Guang-Zhou 510275, P.R. China

Studia Mathematica, Tome 161 (2004) no. 3, pp. 249-268

Résumé

Nonoverlapping 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.

Détail

DOI : 10.4064/sm161-3-3

Keywords: nonoverlapping contractive self similar iterated function systems ifs have studied great detail via set condition other much known about ifs overlaps systems weak separation condition wsc has introduced recently weaker set condition includes many important overlapping cases paper has purposes first consider class self similar measures generated ifs necessary sufficient condition self similar measures absolutely continuous respect hausdorff measures extends result lnr known examples wsc involve algebraic integers golden ratio integral dilation matrices contraction ratios equal second examples self similar ifs wsc arbitrary contraction ratios