Infinite Iterated Function
نویسنده
چکیده
We examine iterated function systems consisting of a countably innnite number of contracting mappings (IIFS). We state results analogous to the well-known case of nitely many mappings (IFS). Moreover, we show that IIFS can be approximated by appropriately chosen IFS both in terms of Hausdorff distance and of Hausdorff dimension. Comparing the descriptive power of IFS and IIFS as mechanisms deening closed and bounded sets, we show that IIFS are strictly more powerful than IFS. On the other hand, there are closed and bounded non-empty sets not de-scribable by IIFS. 1. Introduction and Main Definitions IFS theory, starting out from Hutchinson's paper 14], gained more and more interest. Several books on this topic are available 3, 7, 5, 18, 19] which have become popular even among non-mathematicians. In those references, mostly the case of systems consisting of nitely many mappings has been treated. In this paper, we elaborate the case of countable many mappings, contrasting it with the well-known results on nite IFS. the case of innnitely many mappings has already been considered. We shortly review the main notions and denotations basic to IFS. Throughout this
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