Rigidity of saddle loops
We define an abstract complex saddle loop in C2 as a pair (F,R) of a hyperbolic normalized saddle foliation F with a corner Dulac map D and a regular map R∈Diff(C,0). Up to an appropriate equivalence relation that corresponds to different determinations of complex Dulac and to transversal changes, the first return map is given by F=RD on the universal cover of the standard quadratic domain. We show that such Poincar\' e maps are rigid, in the sense that their non-ramified formal conjugacy implies the analytic conjugacy (in Diff(C,0), lifted to the universal cover).
This is a joint work with D. Panazzolo and L.Teyssier.