幂零流形上路径的渐近发展
The Asymptotic Development of Paths on Nilmanifolds
AI总结：
该研究证明了Carnot尺度下的n步幂零π₁-de Rham定理，通过渐近发展定理建立重标水平路径与幂零发展的收敛关系，扩展了渐近同伦理论的适用范围。
AI中文摘要：
我们证明了一个新的Carnot尺度下的n步幂零π₁-de Rham定理：沿每个收敛尺度序列，长轨迹的渐近同伦类可通过其宏观水平路径的幂零发展来识别。关键分析机制是一个渐近发展定理，表明重标水平路径的一致收敛性结合变差的一致界，可迫使它们完整的Carnot重标幂零发展实现逐层一致收敛。该定理在第一步中重现了Schwartzman的渐近环理论，在第二步中重现了Benardete与Mitchell的2步渐近同伦理论，同时将对应关系扩展到任意幂零步。
英文摘要：
We prove a new Carnot-scale $n$-step nilpotent $π_1$-de Rham theorem: along every convergent scale sequence, the asymptotic homotopy class of a long trajectory is identified with the nilpotent development of its macroscopic horizontal path. The key analytic mechanism is an asymptotic-development theorem showing that uniform convergence of rescaled horizontal paths, together with a uniform bound on variation, forces uniform, layer-by-layer convergence of their full Carnot-rescaled nilpotent developments. The theorem recovers Schwartzman's theory of asymptotic cycles in step one and the 2-step asymptotic homotopy theory of Benardete and Mitchell in step two, while extending the correspondence to arbitrary nilpotent step.
