0 Total convergence or general divergence in Small Divisors

We study generic holomorphic families of dynamical systems presenting problems of small divisors with fixed arithmetic. We prove that we have convergence for all parameter values or divergence everywhere except for an exceptional set of zero Γ-capacity. We illustrate this general principle in different problems of small divisors. As an application we obtain new richer families of non-linearizable examples in the Siegel problem when Bruno condition is violated, generalizing previous results of Yoccoz and the author. In this article we study generic (polynomial) holomorphic families of dynamical systems presenting problems of small divisors with fixed arithmetic. The principle our theorems illustrate is the following : We have total convergence for all parameter values or general divergence except maybe for a very small exceptional set of parameter values. The germinal idea can be traced back to Y. Ilyashenko where in [Il] he studies divergence in problems of small divisors from divergence of the homological (or linearized) equation. Ilyashenko's paper contains a remarkable idea. We find there, for the first time in Small Divisors, the study of linear deformation of the system and the use of the polynomial dependence of the new formal linearizations. A similar idea, but not quite in the same problem, was used by H. Poincaré to show that linear deformations of completely integrable hamiltonians are not generally completely integrable with analytic first integrals depending analytically on the parameter ([Poi] volume I chapter V). It is worth noting that this is the key preliminary step in his difficult proof of the non existence of non trivial local analytic first integrals in the three body problem. Such a linear deformation has been fruitfully used by J.-C. Yoccoz. He proves that in the Siegel problem the quadratic polynomial is the worst linearizable holomorphic germ ([Yo] p. 58). The only ingredient in this proof that is not in Ilyashenko's one is the classical Douady-Hubbard straightening theorem for polynomial-like mappings. Yoccoz simplifies Ilyashenko's argument replacing Nadirashvili's lemma by the maximum principle.