复杂极限环的快速演化
Rapid evolution of complex limit cycles
摘要：
本文研究了复平面上具有奇点的全纯叶理的复圈的若干问题。我们专注于当多项式微分的一种形式产生的黎曼曲面的叶理的情况下。在这种情况下，一个复杂的循环被定义为一个非平凡的元素的基本群的叶子从叶理。每当多项式叶理来自于一个精确的1-形式的扰动时，人们可以引入多重循环的概念。这种类型的循环至少有一个代表，确定一个自由的同伦类的循环在一个开放的纤维子域的复平面。这个子域的拓扑结构与前面提到的精确的一形式密切相关。本文引入并研究了近于可积多项式叶理的多重圈的概念。我们还探讨了这些周期如何对应于周期轨道的某个庞加莱映射与叶理。最后，我们讨论了连续族多重极限环在复平面中当叶理收敛到其可积部分时从某些大开域逃逸的趋势。
英文摘要：
The current article studies certain problems related to complex cycles of holomorphic foliations with singularities in the complex plane. We focus on the case when polynomial differential one-form gives rise to a foliation by Riemann surfaces. In this setting, a complex cycle is defined as a nontrivial element of the fundamental group of a leaf from the foliation. Whenever the polynomial foliation comes from a perturbation of an exact one-form, one can introduce the notion of a multi-fold cycle. This type of cycle has at least one representative that determines a free homotopy class of loops in an open fibered subdomain of the complex plane. The topology of this subdomain is closely related to the exact one-form mentioned earlier. We introduce and study the notion of multi-fold cycles of a close-to-integrable polynomial foliation. We also explore how these cycles correspond to periodic orbits of a certain Poincaré map associated with the foliation. Finally, we discuss the tendency of a continuous family of multi-fold limit cycles to escape from certain large open domains in the complex plane as the foliation converges to its integrable part.
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