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Transactions of the American Mathematical Society

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


Author: Adrian Jenkins
Journal: Trans. Amer. Math. Soc. 360 (2008), 331-346
MSC (2000): Primary 30D05
DOI: https://doi.org/10.1090/S0002-9947-07-04266-3
Published electronically: May 16, 2007
MathSciNet review: 2342005
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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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Additional Information

Adrian Jenkins
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Address at time of publication: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: majenkin@math.purdue.edu

DOI: https://doi.org/10.1090/S0002-9947-07-04266-3
Keywords: Smooth conjugacy, holomorphic germ, time-one map
Received by editor(s): May 18, 2005
Received by editor(s) in revised form: February 7, 2006
Published electronically: May 16, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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