The moduli space of germs of generic families of analytic diffeomorphisms unfolding of a codimension one resonant diffeomorphism or resonant saddle
This interesting paper completes previous works by the author and C. Christopher giving a complete description of the moduli space under holomorphic conjugacy for germs of generic 1-parameter families unfolding a 1-dimensional resonant biholomorphism or a 2-dimensional resonant saddles. This is obtained in several stages. First of all, the author shows that any germ of a family unfolding a resonant biholomorphism can be realized as the holonomy map of a separatrix of a germ of family of vector fields having a second separatrix whose family of holonomies is again a germ of family of resonant biholomorphisms. Iterating this trick a finite number of times allows the author to reduce the problem to the study of the moduli space of families unfolding a germ of biholomorphism tangent to the identity and with a double fixed point at the origin. This requires relating the (Écalle, Martinet-Ramis, and/or Glutsyuk) invariants associated to the holonomies of the two separatrices in a germ of family unfolding a resonant saddle; the author, mostly using the Martinet-Ramis point of view, is able to give exact formulas for the relationships between the formal and the analytic part of the two invariants. The invariants of a family (in prepared form) unfolding a germ tangent to the identity with a double fixed point are: the formal invariant \(a(\varepsilon)\), depending analytically on the parameter \(\varepsilon\) of the unfolding; and the analytical invariant given by two families of germs of biholomorphisms \((\psi^0_{\varepsilon,\pm},\psi^\infty_{\varepsilon,\pm})|_{\varepsilon\in V_{\pm}}\), where \(V_{\pm}\) are two sectors whose union is a punctured neighborhood of the origin in the \(\varepsilon\)-space, and \(\psi^0_{\varepsilon,\pm}\) (respectively, \(\psi^\infty_{\varepsilon,\pm}\)) is a germ of biholomorphism at the origin (respectively, at infinity), satisfying a suitable compatibility condition implying that, possibly after a linear change of coordinates in \(\mathbb{CP}^1\), they are 1-summable in~\(\varepsilon\). The author is then able to prove that every pair of a formal and an analytical invariant is realizable by a family unfolding a germ tangent to the identity. Putting everything together, this yields a realization theorem for the moduli spaces of unfoldings of resonant biholomorphisms and of unfoldings of resonant saddles. Finally, the author also studies the particular case of an unfolding of a resonant real vector field.
scientific article; zbMATH DE number 5687483
scientific article