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Parabolic curves for diffeomorphisms in \(\mathbb C^2\). (English) Zbl 1154.32005

It is known after [J. Écalle, Publ. Math. Orsay 85-05 (1985; Zbl 0602.30029)] and [M. Hakim, Duke Math. J. 92, 403–428 (1998; Zbl 0952.32012)] that a “generic” germ of biholomorphism at \(O\) in \(\mathbb C^n\) tangent to the identity has “petals”, namely, it posses invariant analytic discs containing \(O\) on the boundary and such that its dynamics along those discs is conjugated to parabolic dynamics. Later M. Abate [Duke Math. J. 107, No. 1, 173–207 (2001; Zbl 1015.37035)] proved that on \(\mathbb C^2\) every germ tangent to the identity posses petals. A similar result holds for germs of holomorphic vector fields: C. Camacho and P. Sad [Ann. Math. (2) 115, 579–595 (1982; Zbl 0503.32007)] proved that every germ of holomorphic vector field in \(\mathbb C^2\) admits a separatrix. The proof of both Abate’s and Camacho-Sad’s results relies on a sequences of blowing ups reducing the singularities which can be suitably selected by means of an index theorem. Later the reviewer [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 2, No. 3, 493–520 (2003; Zbl 1170.32311)] found a way to obtain (and generalize to some singular setting) Abate’s result by using a first-jet version of the Camacho-Sad approach.
In the paper under review, the authors give a proof of Abate’s theorem using a formal version of the Camacho-Sad result. Namely, they consider the formal vector field whose time one flow is the given germ of biholomorphism and apply Camacho-Sad’s result to such a formal germ. Then, they come up with a “good” singular point for the vector field which is a “generic” point for the germ of biholomorphism where one can apply Écalle-Hakim’s result and blow down the petals.

MSC:

32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37F99 Dynamical systems over complex numbers
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