Porosities of the sets of attractors

TL;DR

This paper investigates the porosity properties of attractor sets for iterated function systems and weak iterated function systems on [0,1], revealing that the set of attractors for standard systems is not strongly porous and that the difference with weak attractors is dense.

Contribution

It demonstrates that the family of attractors for standard IFSs is not $ ext{sigma}$-strongly porous and that weak attractors outside this family are dense in the hyperspace.

Findings

$A[0,1]$ is not $ ext{sigma}$-strongly porous.

$A_w[0,1]ackslash A[0,1]$ is dense in $K([0,1])$.

Both sets are meager in the hyperspace.

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 $K([0,1])$.