AI 中文总结
本文通过模板比较Lorenz、马蹄、Rössler系统的周期轨道类型，证明二者存在非平凡重叠，同时构造无限多无法嵌入马蹄模板的Lorenz链接，明确了Lorenz链接与马蹄/Rössler链接的差异。
AI 中文摘要
我们通过Lorenz、马蹄（horshoe）和Rössler系统对应的模板，比较它们的周期轨道类型。Lorenz链接由Lorenz模板承载，马蹄链接由马蹄模板承载；对于本文考虑的标准模板模型，Rössler模板在反演对称下与马蹄机制的模板等价，因此本文将Rössler链接和马蹄链接归为同一模板类。我们证明Lorenz链接与马蹄/Rössler链接的关系具有两个互补方面：其一，通过构造可嵌入两个模板的无限族链接，证明两类链接存在非平凡重叠，验证了Kofman提出的“马蹄链接也应为Lorenz链接”的猜想；其二，两类链接并非完全相同，Holmes和Williams已证明许多Lorenz环面纽结无法嵌入马蹄模板：若环面纽结T(p,q)（p<q）为马蹄纽结，则满足3p≤2q，我们将Holmes-Williams的这一障碍从环面纽结扩展至环面链接，并构造了无限多个双曲Lorenz纽结与链接，以及无限多个卫星Lorenz纽结与链接，这些均无法嵌入马蹄模板，因此属于非马蹄链接、进而非Rössler链接的Lorenz链接。
英文摘要
We compare the periodic orbit types of the Lorenz, horseshoe, and Rössler systems through their associated templates. Lorenz links are carried by the Lorenz template, while horseshoe links are carried by the horseshoe template. For the standard template model considered here, the Rössler template is identified, up to inversion symmetry, with the template of the horseshoe mechanism. Thus, in this paper, Rössler links and horseshoe links are treated as belonging to the same template class. We show that the relationship between Lorenz links and horseshoe/Rössler links has two complementary sides. First, we prove that the overlap between the two families is nontrivial by constructing infinite families of links which can be embedded in both templates. This verifies, for these families, a conjecture stated by Kofman that horseshoe links should also be Lorenz links. On the other hand, we prove that the two families are far from being the same. Holmes and Williams showed that many Lorenz torus knots cannot be embedded in the horseshoe template: if the torus knot \(T(p,q)\), with \(p<q\), is a horseshoe knot, then \(3p\leq 2q\). We show that this phenomenon is much broader. We extend the Holmes--Williams obstruction from torus knots to torus links, and we construct infinitely many hyperbolic Lorenz knots and links, as well as infinitely many satellite Lorenz knots and links, which cannot be embedded in the horseshoe template. Consequently, these examples are Lorenz links which are not horseshoe links and hence not Rössler links.
Comments26 pages, 2 figures
