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arXiv · 1905.13507

On a typical compact set as the attractor of generalized iterated function systems of infinite order

Łukasz Maślanka
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Abstract

In 2013 Balka and Máthé showed that in uncountable polish spaces the typical compact set is not a fractal of any IFS. In 2008 Miculescu and Mihail introduced a concept of a generalized iterated function system (GIFS in short), a particular extension of classical IFS, in which they considered families of mappings defined on finite Cartesian product $X^m$ with values in $X$. Recently, Secelean extended these considerations to mappings defined on the space $\ell_\infty(X)$ of all bounded sequences of elements of $X$ endowed with supremum metric. In the paper we show that in Euclidean spaces a typical compact set is an attractor in sense of Secelean and that in general in the polish spaces it can be perceived as selfsimilar in such sense.

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Łukasz Maślanka. 2019-05-31. On a typical compact set as the attractor of generalized iterated function systems of infinite order. https://arxiv.org/abs/1905.13507

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