PreprintCC BY 4.0arXiv
The circle as a topological fractal
Abstract
We prove that no family of two continuous self-maps witnesses that the circle is a topological fractal, answering a question of Karasová and the present author. Since three maps are known to suffice, this bound is optimal. In contrast, for every ε>0 there are two continuous self-maps of the circle, depending on ε, whose images cover the circle and an integer N such that every composition of N of them has image of diameter less than ε. Thus two maps suffice at any prescribed scale, but no fixed pair works at all scales.
Related
BibTeX
@misc{vejnar2026circle,
title = {{The circle as a topological fractal}},
author = {Benjamin Vejnar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21611},
eprint = {2609.21611},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21611},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The circle as a topological fractal
AU - Benjamin Vejnar
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21611
UR - https://arxiv.org/abs/2609.21611
AB - We prove that no family of two continuous self-maps witnesses that the circle is a topological fractal, answering a question of Karasová and the present author. Since three maps are known to suffice, this bound is optimal. In contrast, for every ε>0 there are two continuous self-maps of the circle, depending on ε, whose images cover the circle and an integer N such that every composition of N of them has image of diameter less than ε. Thus two maps suffice at any prescribed scale, but no fixed pair works at all scales.
ER -