2011/06/14 by Nikolay Dimitrov, Dimitrov, Nikolay
Mathematics · #37C27 (Primary) #37F75 #37G15 #55R10 #57M10 (Secondary) #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #math.CV #math.DS #math.GT #msc:37C27 #msc:37F75 #msc:37G15 #msc:55R10 #msc:57M10
paper · pdf · doi:10.48550/arxiv.1106.2786
27 pages, submitted to "Discrete and Continuous Dynamical Systems" - Series A
arxiv created 2011/06/14 · arxiv updated 2011/06/15
In the current article we study complex cycles of higher multiplicity in a specific polynomial family of holomorphic foliations in the complex plane. The family in question is a perturbation of an exact polynomial one-form giving 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. In addition to that, we introduce the notion of a multi-fold cycle and show that in our example there exists a limit cycle of any multiplicity. Furthermore, such a cycle gives rise to a one-parameter family of cycles continuously depending on the perturbation parameter. As the parameter decreases in absolute value, the cycles from the continuous family escape from a very large subdomain of the complex plane.