arXiv subjects
Publications and source records attributed to Paweł Klinga.
We investigate the $σ$-porosity of certain known ideals of subsets of natural numbers. Porosity is a notion of smallness in metric spaces that is stronger than nowhere density. Analogously, $σ$-porosity is a strengthening of meagerness. In this paper, we verify which ideals are $σ$-porous.
This paper is an attempt to measure the difference between the family of iterated function systems attractors and a broader family, the set of attractors for weak iterated function systems. We discuss Borel complexity of the set wIFS$^d$ of attractors for weak iterated function systems acting on $[0,1]^d$ (as a subset of the hyperspace $K([0,1]^d)$ of all compact subsets of $[0,1]^d$ equipped in the Hausdorff metric). We prove that wIFS$^d$ is $G_{δσ}$-hard in $K([0,1]^d)$, for all $d\in\mathbb{N}$. In particular, wIFS$^d$ is not $F_{σδ}$ (in contrast to the family IFS$^d$ of attractors for classical iterated function systems acting on $[0,1]^d$, which is $F_σ$). Moreover, we show that in the one-dimensional case, wIFS$^1$ is an analytic subset of $K([0,1])$.
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 $K([0,1])$.
For $n,d\in\mathbb{N}$ we consider the families: - $L_n^d$ of attractors for iterated function systems (IFS) consisting of $n$ contractions acting on $[0,1]^d$, - $wL_n^d$ of attractors for weak iterated function systems (wIFS) consisting of $n$ weak contractions acting on $[0,1]^d$. We study closures of the above families as subsets of the hyperspace $K([0,1]^d)$ of all compact subsets of $[0,1]^d$ equipped in the Hausdorff metric. In particular, we show that $\overline{L_n^d}=\overline{wL_n^d}$ and $L_{n+1}^d\setminus\overline{L_n^d}\neq\emptyset$, for all $n,d\in\mathbb{N}$. What is more, we construct a compact set belonging to $\overline{L_2^d}$ which is not an attractor for any wIFS. We present a diagram summarizing our considerations.