We study the formal conjugacy properties of germs of complex analytic diffeomorphisms defined in the neighborhood of the origin of ${ \mathbb C}^{ n}$. More precisely, we are interested on the nature of formal conjugations along the fixed points set. We prove that there are formally conjugated local diffeomorphisms $\phi, \eta$ such that every formal conjugation $\hat{ \sigma}$ (i.e. $\eta \circ \hat{ \sigma} = \hat{ \sigma} \circ \phi$) does not extend to the fixed points set $Fix (\phi)$ of $\phi$, meaning that it is not transversally formal (or semi-convergent) along $Fix (\phi)$. 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.
Categories: math.DS math.CV