Big Crunch on Julia and Anti-Julia Sets in the Integrable Limit
Abstract
The Julia set is defined by the closure of repelling periodic points in chaotic systems. Why does this structure not appear in integrable systems? In this paper, we address this question by demonstrating the existence of the closure of divergences of the periodic equations, which we designate as the "anti-Julia set." We also call the sets before taking the closures the pre-Julia and pre-anti-Julia sets, respectively. We illustrate the transition mechanism by considering a complex map that interpolates between integrable and non-integrable dynamics, by introducing a real deformation parameter a. For 0<a≤ 1/2, we show that the Julia set and the anti-Julia set coincide in the complex plane, although the pre-Julia and pre-anti-Julia sets remain completely disjoint. At the integrable limit a→ 0, these two dual structures undergo a critical collision and subsequent annihilation, reminiscent of a cosmological Big Crunch. When 1/2<a<1, on the other hand, the boundary of the pre-anti-Julia set includes the Julia set. We analytically characterize these phenomena, focusing in particular on the asymptotic behavior of the pre-anti-Julia set as it approaches the integrable limit, and provide numerical visualizations that elucidate the underlying mechanisms of this Big Crunch phenomenon.
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BibTeX
@misc{iizawa2026crunch,
title = {{Big Crunch on Julia and Anti-Julia Sets in the Integrable Limit}},
author = {Masatomi Iizawa and Satoru Saito and Yasuhito Narita and Yasuyuki Maeda and Hiromitsu Harada and Akira Shudo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07308},
eprint = {2609.07308},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07308},
}
RIS
TY - UNPB
TI - Big Crunch on Julia and Anti-Julia Sets in the Integrable Limit
AU - Masatomi Iizawa
AU - Satoru Saito
AU - Yasuhito Narita
AU - Yasuyuki Maeda
AU - Hiromitsu Harada
AU - Akira Shudo
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07308
UR - https://arxiv.org/abs/2609.07308
AB - The Julia set is defined by the closure of repelling periodic points in chaotic systems. Why does this structure not appear in integrable systems? In this paper, we address this question by demonstrating the existence of the closure of divergences of the periodic equations, which we designate as the "anti-Julia set." We also call the sets before taking the closures the pre-Julia and pre-anti-Julia sets, respectively. We illustrate the transition mechanism by considering a complex map that interpolates between integrable and non-integrable dynamics, by introducing a real deformation parameter a. For 0<a≤ 1/2, we show that the Julia set and the anti-Julia set coincide in the complex plane, although the pre-Julia and pre-anti-Julia sets remain completely disjoint. At the integrable limit a→ 0, these two dual structures undergo a critical collision and subsequent annihilation, reminiscent of a cosmological Big Crunch. When 1/2<a<1, on the other hand, the boundary of the pre-anti-Julia set includes the Julia set. We analytically characterize these phenomena, focusing in particular on the asymptotic behavior of the pre-anti-Julia set as it approaches the integrable limit, and provide numerical visualizations that elucidate the underlying mechanisms of this Big Crunch phenomenon.
ER -