Extension of formal conjugations between diffeomorphisms

We study the formal conjugacy properties of germs of complex analytic diffeomorphisms defined in the neighborhood of the origin of ℂ n . More precisely, we are interested in the nature of formal conjugations along the fixed points set. We prove that there are formally conjugated local diffeomorphisms φ, η such that every formal conjugation (i.e. ) does not extend to the fixed points set Fix ( φ ) of φ , meaning that it is not transversally formal (or semi-convergent) along Fix ( φ ). We focus on unfoldings of 1-dimensional tangent to the identity diffeomorphisms. We identify the geometrical configurations preventing formal conjugations to extend to the fixed points set: roughly speaking, either the unperturbed fiber is singular or generic fibers contain multiple fixed points.