发表机构
Institute of Science and Technology Austria; Universitat Politècnica de Catalunya; Centre de Recerca Matemàtica（奥地利科学技术研究所; 加泰罗尼亚理工大学; 数学研究中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文受Arnold启发，证明受限平面圆型(n+1)体问题的庞加莱映射重整化可任意精度逼近任意辛嵌入，从而给出天体力学中普适动力学的弱有限维形式。
AI 中文摘要
普适映射（其重整化迭代可逼近给定类中任意映射的映射）在若干微分同胚空间中是局部通有的[Bonatti--Díaz 2003, Turaev 2015]。在流体动力学中，庞加莱映射为普适映射的稳态欧拉流也被证明是局部稠密的[Berger--Florio--PeraltaSalas, 2023]。受Arnold关于天体力学中轨道复杂度与流体中轨道复杂度应可比的远见[Arnold, 1966]的启发，本文探讨了天体力学中是否存在普适映射的问题。我们通过证明一种弱化的、有限维形式的普适动力学给出了部分回答。更具体地，我们证明：对于适当的n和主天体质量的适当选择，圆盘到R^2的任意辛嵌入都可以以任意精度被受限平面圆型(n+1)体问题的庞加莱映射的重整化所逼近。证明依赖于Gonchenko--Shilnikov--Turaev理论[Turaev 2003, Gonchenko--Turaev--Shilnikov 2007]，并需要控制任意高阶同宿切点附近的动力学；[Garrido--Martín--Paradela, 2025]中发展的技术对此控制至关重要。
英文摘要
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 $\mathbb{R}^2$ 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.
