Porosities of the sets of attractors
Paweł Klinga, Adam Kwela
TL;DR
The paper investigates two attractor families, $A[0,1]^d$ and $A_w[0,1]^d$, as subsets of the hyperspace $K([0,1]^d)$. It develops the hyperspace and porosity framework, showing that $A[0,1]^d$ is not $\sigma$-strongly porous in $K([0,1]^d)$ while the difference $A_w[0,1]^d\setminus A[0,1]^d$ is dense in $K([0,1]^d)$. A dimension-raising transfer is used to connect attractors across dimensions. The core contribution is a coding-based argument for the non-$\sigma$-strong-porosity of classical attractors, complemented by a density result for weak-attractor deviations, clarifying structural distinctions between IFS and wIFS attractor spaces. These results advance the understanding of the geometric and descriptive-set-theoretic structure of attractor families in hyperspaces of compact sets.
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 $σ$-lower porous while the question about $σ$-lower porosity of $A_w[0,1]$ is still open. We prove that $A[0,1]$ is not $σ$-strongly porous in $K([0,1])$. Moreover, we show that $A_w[0,1]\setminus A[0,1]$ is dense in $K([0,1])$.