Universal Dynamics in a family of Celestial Mechanics models
Abstract
Universal maps (maps whose renormalized iterations approximate every map in a given class) are locally generic in several spaces of diffeomorphisms [Bonatti--Díaz 2003, Turaev 2015]. In fluid dynamics, steady Euler flows whose Poincaré maps are universal are also known to be locally dense [Berger--Florio--PeraltaSalas, 2023]. Motivated by Arnold's vision [Arnold, 1966] that the complexity of orbits in celestial mechanics and that in fluids should be comparable, in the present paper we address the question of the existence of universal maps in celestial mechanics. We give a partial answer by proving a weak, finite-dimensional form of universal dynamics. More concretely, we show that every symplectic embedding of the disk into ℝ2 can be approximated, with arbitrary precision, by a renormalization of a Poincaré map of a restricted planar circular (n+1)-body problem, for suitable n and appropriate choice of the masses of the primaries. The proof relies on Gonchenko--Shilnikov--Turaev theory [Turaev 2003, Gonchenko--Turaev--Shilnikov 2007] and requires control of the dynamics near homoclinic tangencies of arbitrarily high order; the techniques developed in [Garrido--Martín--Paradela, 2025] are essential for such control.
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BibTeX
@misc{garrido2026universal,
title = {{Universal Dynamics in a family of Celestial Mechanics models}},
author = {Miguel Garrido and Pau Martín},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09966},
eprint = {2610.09966},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09966},
}
RIS
TY - UNPB
TI - Universal Dynamics in a family of Celestial Mechanics models
AU - Miguel Garrido
AU - Pau Martín
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09966
UR - https://arxiv.org/abs/2610.09966
AB - Universal maps (maps whose renormalized iterations approximate every map in a given class) are locally generic in several spaces of diffeomorphisms [Bonatti--Díaz 2003, Turaev 2015]. In fluid dynamics, steady Euler flows whose Poincaré maps are universal are also known to be locally dense [Berger--Florio--PeraltaSalas, 2023]. Motivated by Arnold's vision [Arnold, 1966] that the complexity of orbits in celestial mechanics and that in fluids should be comparable, in the present paper we address the question of the existence of universal maps in celestial mechanics. We give a partial answer by proving a weak, finite-dimensional form of universal dynamics. More concretely, we show that every symplectic embedding of the disk into ℝ^2 can be approximated, with arbitrary precision, by a renormalization of a Poincaré map of a restricted planar circular (n+1)-body problem, for suitable n and appropriate choice of the masses of the primaries. The proof relies on Gonchenko--Shilnikov--Turaev theory [Turaev 2003, Gonchenko--Turaev--Shilnikov 2007] and requires control of the dynamics near homoclinic tangencies of arbitrarily high order; the techniques developed in [Garrido--Martín--Paradela, 2025] are essential for such control.
ER -