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Separation Conditions for Iterated Function Systems with a Common Attractor
Abstract
Let K be the non-singleton attractor of a finite iterated function system Φ={ϕi(x)=ai Oi x+ti}i=1m on ℝd, where 0<ai<1, ti∈ℝd and the Oi are orthogonal transformations on ℝd. Suppose that Φ satisfies the open set condition, but not the strong separation condition. We show that K cannot be generated by any finite iterated function system of similitudes satisfying the strong separation condition. This removes the homogeneity assumption from a result of Feng, Ruan and Xiong and gives a negative answer to a folklore question about the separation conditions on the generating iterated function systems of self-similar sets.
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BibTeX
@misc{ma2026separation,
title = {{Separation Conditions for Iterated Function Systems with a Common Attractor}},
author = {Cai-Yun Ma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17190},
eprint = {2609.17190},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17190},
}
RIS
TY - UNPB
TI - Separation Conditions for Iterated Function Systems with a Common Attractor
AU - Cai-Yun Ma
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17190
UR - https://arxiv.org/abs/2609.17190
AB - Let K be the non-singleton attractor of a finite iterated function system Φ={ϕ_i(x)=a_i O_i x+t_i}_i=1^m on ℝ^d, where 0<a_i<1, t_i∈ℝ^d and the O_i are orthogonal transformations on ℝ^d. Suppose that Φ satisfies the open set condition, but not the strong separation condition. We show that K cannot be generated by any finite iterated function system of similitudes satisfying the strong separation condition. This removes the homogeneity assumption from a result of Feng, Ruan and Xiong and gives a negative answer to a folklore question about the separation conditions on the generating iterated function systems of self-similar sets.
ER -