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arXiv · 1104.3912

Extension of formal conjugations between diffeomorphisms

Javier Ribón
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Abstract

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 $ϕ, η$ such that every formal conjugation $\hatσ$ (i.e. $η\circ \hatσ = \hatσ \circ ϕ$) 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.

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Javier Ribón. 2011-04-20. Extension of formal conjugations between diffeomorphisms. https://doi.org/10.1007/s00574-012-0012-4

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