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Mihail and Miculescu introduced the notion of a generalized iterated function system (GIFS in short), and proved that every GIFS generates an attractor. (In our previous paper we gave this notion a more general setting.) In this paper we show that for any m≥2, there exists a Cantor subset of the plane which is an attractor of some GIFS of order m , but is not an attractor of a GIFS of order m−1. In particular, this result shows that there is a subset of the plane which is an attractor of some GIFS, but is not an attractor of an IFS. We also give an example of a Cantor set which is not an attractor of a GIFS. 相似文献
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