Borel complexity of the family of attractors for weak IFSs
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910770654085120 |
|---|---|
| author | Klinga, Paweł Kwela, Adam |
| author_facet | Klinga, Paweł Kwela, Adam |
| contents | 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 $K([0,1])$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_01921 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Borel complexity of the family of attractors for weak IFSs Klinga, Paweł Kwela, Adam Dynamical Systems General Topology 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 $K([0,1])$. |
| title | Borel complexity of the family of attractors for weak IFSs |
| topic | Dynamical Systems General Topology |
| url | https://arxiv.org/abs/2111.01921 |