2010/05/11 by Nikolay Dimitrov, Dimitrov, Nikolay
Mathematics · #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #math.CV #math.DS #math.GT
paper · pdf · doi:10.48550/arxiv.1005.1697
44 pages
arxiv created 2010/05/11 · arxiv updated 2010/05/12
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.