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

## TL;DR

This paper demonstrates that in Euclidean and polish spaces, typical compact sets can be viewed as attractors of generalized iterated function systems, extending classical fractal concepts to broader contexts.

## Contribution

It shows that typical compact sets in Euclidean and polish spaces are attractors of generalized IFS, broadening the understanding of fractal structures in these spaces.

## Key findings

- Typical compact sets in Euclidean spaces are attractors of Secelean's generalized IFS.
- In polish spaces, typical compact sets can be perceived as selfsimilar attractors.
- Extends classical IFS theory to generalized systems in infinite-dimensional contexts.

## Abstract

In 2013 Balka and M\'ath\'e 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.