Holomorphic germs and the problem of smooth conjugacy in a punctured neighborhood of the origin

Jenkins, Adrian

doi:10.1090/S0002-9947-07-04266-3

Holomorphic germs and the problem of smooth conjugacy in a punctured neighborhood of the origin
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Trans. Amer. Math. Soc. 360 (2008), 331-346 Request permission

Abstract:

We consider germs of conformal mappings tangent to the identity at the origin in $\mathbf {C}$. We construct a germ of a homeomorphism which is a $C^{\infty }$ diffeomorphism except at the origin conjugating these holomorphic germs with the time-one map of the vector field $V(z)=z^{m}\tfrac {\partial }{\partial z}$. We then show that, in the case $m=2$, for a germ of a homeomorphism which is real-analytic in a punctured neighborhood of the origin, with real-analytic inverse, conjugating these germs with the time-one map of the vector field exists if and only if a germ of a biholomorphism exists.

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