Abstract. 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 Γcapacity. 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
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