Dynamics of multi-resonant biholomorphisms (Q2878677)

The authors study the dynamics of \(F\), a germ of biholomorphism of \(\mathbb C^n\) at the origin, whose linear part is diagonalizable. They define \(F\) to be multi-resonant with respect to the first \(r\) eigenvalues of the linear part if the resonances between these eigenvalues are finitely generated, with \(m\) generators. Given such a biholomorphism \(F\) and a choice of the Poincaré-Dulac normal form, one can define a monomial map \(\pi: \mathbb C^n \to \mathbb C^m\) that semi-conjugates \(F\) to a germ \(f\) of biholomorphisms of \(\mathbb C^m\) tangent to the identity, called the parabolic shadow of \(F\). Using Hakim's theory on germs tangent to the identity, and under some natural conditions on \(f\), and on the dynamics of \(F\) on the fibers of \(\pi\), the authors are able to prove that the initial germ \(F\) admits a basin of attraction (and in fact at least as many as the weighted order of \(F\)).