Parabolic curves of diffeomorphisms asymptotic to formal invariant curves. (English) Zbl 1412.32015
A parabolic curve for a holomorphic germ \(f\) tangent to the identity at the origin in several complex variables is a one-dimensional holomorphic curve (with the origin in its boundary) which is \(f\)-invariant and where the dynamics reproduces the parabolic one-dimensional dynamics of a petal in the classical Leau-Fatou flower theorem. J. Écalle [Publ. Math. Orsay 85–05, 585 p. (1985; Zbl 0602.30029)] and, with different methods, M. Hakim [Duke Math. J. 92, No. 2, 403–428 (1998; Zbl 0952.32012)] proved the existence of parabolic curves tangent to the so-called non-degenerate characteristic directions, and that if a parabolic curve is tangent at the origin to some direction then that direction must be characteristic. A germ tangent to the identity always have characteristic directions, but they might all be degenerate; so it remains open the problem of existence of parabolic curves tangent to a given degenerate characteristic direction.
This paper deals with the closely related problem of existence of a parabolic curve asymptotic to a (necessarily invariant) formal curve. In other words, instead of prescribing only the first tangent (the characteristic direction) of a parabolic curve all higher-order tangencies are prescribed too. This is relevant because 2-dimensional holomorphic germs tangent to the identity always admit a formal invariant curve (necessarily tangent to a possibly degenerate characteristic direction).
The main result of this paper says that, in dimension 2, given a formal invariant curve not contained in the set of fixed points of a germ tangent to the identity \(f\) then there always exists a parabolic curve for \(f\) or \(f^{-1}\) which is asymptotic to the formal curve. The proof consists of two steps. First, a reduction of the pair (germ, invariant formal curve) to a normal form by means of change of coordinates and blow-ups; then when the pair is in normal form the authors prove the existence of a parabolic curve for the germ (or its inverse) asymptotic to the formal curve, adapting Hakim’s arguments.
Finally, in the last section of the paper the authors outline a way to generalise their approach to germs tangent to the identity in \(\mathbb{C}^n\) with \(n\ge 3\).
This paper deals with the closely related problem of existence of a parabolic curve asymptotic to a (necessarily invariant) formal curve. In other words, instead of prescribing only the first tangent (the characteristic direction) of a parabolic curve all higher-order tangencies are prescribed too. This is relevant because 2-dimensional holomorphic germs tangent to the identity always admit a formal invariant curve (necessarily tangent to a possibly degenerate characteristic direction).
The main result of this paper says that, in dimension 2, given a formal invariant curve not contained in the set of fixed points of a germ tangent to the identity \(f\) then there always exists a parabolic curve for \(f\) or \(f^{-1}\) which is asymptotic to the formal curve. The proof consists of two steps. First, a reduction of the pair (germ, invariant formal curve) to a normal form by means of change of coordinates and blow-ups; then when the pair is in normal form the authors prove the existence of a parabolic curve for the germ (or its inverse) asymptotic to the formal curve, adapting Hakim’s arguments.
Finally, in the last section of the paper the authors outline a way to generalise their approach to germs tangent to the identity in \(\mathbb{C}^n\) with \(n\ge 3\).
Reviewer: Marco Abate (Pisa)
MSC:
| 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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