Porosities of the sets of attractors

Paweł Klinga; Adam Kwela

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arXiv · 2111.01927

Porosities of the sets of attractors

Paweł Klinga·Adam Kwela

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Abstract

This paper is another attempt to measure the difference between the family $A[0,1]$ of attractors for iterated function systems acting on $[0,1]$ and a broader family, the set $A_w[0,1]$ of attractors for weak iterated function systems acting on $[0,1]$. It is known that both $A[0,1]$ and $A_w[0,1]$ are meager subsets of the hyperspace $K([0,1])$ (of all compact subsets of $[0,1]$ equipped in the Hausdorff metric). Actually, $A[0,1]$ is even $\sigma$-lower porous while the question about $\sigma$-lower porosity of $A_w[0,1]$ is still open. We prove that $A[0,1]$ is not $\sigma$-strongly porous in $K([0,1])$. Moreover, we show that $A_w[0,1]\setminus A[0,1]$ is dense in $K([0,1])$.

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Paweł Klinga, Adam Kwela. 2021-11-02. Porosities of the sets of attractors. https://arxiv.org/abs/2111.01927

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