Complex length and persistence of limit cycles in a neighborhood of a hyperbolic polycycle (Q482177)

The paper under review deals with persistence problems of complex limit cycles in one-parameter analytic families of holomorphic foliations in \(\mathbb C^2\). The author considers the case of an initial foliation \(\mathcal F_0\) with a polycycle whose vertexes are diophantine singular points, and a deformation which preserves the diophantine character of the vertexes and where saddle connections between two singular points are isolated. Under these conditions, he shows that any complex limit cycle of the foliation \(\mathcal F_0\) in a neighborhood of the polycyle either changes holomorphically under the change of the parameter or goes to the boundary of the fixed neighborhood of the polycyle.