Compensation of small denominators and ramified linearisation of local objects.

*(English)*Zbl 0810.58036
Camacho, C. (ed.) et al., Complex analytic methods in dynamical systems. Proceedings of the congress held at Instituto de Matemática Pura e Aplicada, IMPA, Rio de Janeiro, Brazil, January 1992. Paris: Société Mathématique de France, Astérisque. 222, 135-199 (1994).

In the analytic linearization of a (local) analytic vector field or a (local) analytic diffeomorphism at 0 in \(\mathbb{C}^ \nu\), the scalars of the linear parts (after applying the preparation theorem) play a fundamental role in finding interesting properties of the field or the diffeo. There are three interesting complicating situations: resonance, quasiresonance, and nihilence (the existence of a power series first integral and small denominators). Main result: If the vector field or diffeo is formal and nondegenerate, then it can be linearized via a so- called formal-ramified change of coordinates; if it is analytic and nondegenerate, then it can be linearized via a so-called seriable- ramified change of coordinates. These statements do not depend on whether the local object is resonant or not. “Seriable-ramified” roughly means that one has a formal change of coordinates with a bounded lim sup condition on the exponents which can be resummed in a unique way using a suitable Borel-Laplace summation procedure to a sum that is defined and analytic in a ramified spiralling neighborhood of the origin. The idea of seriable linearization rests on the idea of “compensators”, i.e., sums of monomials that remain small although their coefficients may be very large (due to internal cancellations).

The author proves the result stated above, for the important subclass of girators (i.e., vector fields with purely imaginary eigenvalues) and girations (diffeomorphisms with eigenvalues of modulus 1). This is to be extended to more general local objects in a later paper. The main purpose is the development of seriable linearization: this property is quasianalytic in a certain sense, it holds in all cases (even in the presence of resonance), the domain of seriable linearization depends continuously on the multipliers and is always fairly large (unlike the Siegel linearization domain in the case of analytic linearization), the optimal spiralling speed can be explicitly determined, there is an interrelation between seriability and compensated resurgence, and the notion can be extended to non-analytic local objects (e.g., defined by Gevrey power series).

This paper is very technical and is written in the author’s usual abstract terminology.

For the entire collection see [Zbl 0797.00019].

The author proves the result stated above, for the important subclass of girators (i.e., vector fields with purely imaginary eigenvalues) and girations (diffeomorphisms with eigenvalues of modulus 1). This is to be extended to more general local objects in a later paper. The main purpose is the development of seriable linearization: this property is quasianalytic in a certain sense, it holds in all cases (even in the presence of resonance), the domain of seriable linearization depends continuously on the multipliers and is always fairly large (unlike the Siegel linearization domain in the case of analytic linearization), the optimal spiralling speed can be explicitly determined, there is an interrelation between seriability and compensated resurgence, and the notion can be extended to non-analytic local objects (e.g., defined by Gevrey power series).

This paper is very technical and is written in the author’s usual abstract terminology.

For the entire collection see [Zbl 0797.00019].

Reviewer: J.S.Joel (Kelly)

##### MSC:

37G05 | Normal forms for dynamical systems |

32S65 | Singularities of holomorphic vector fields and foliations |

32A30 | Other generalizations of function theory of one complex variable (should also be assigned at least one classification number from Section 30-XX) |

30B99 | Series expansions of functions of one complex variable |