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Quasi-conformal deformations of nonlinearizable germs

Author: Kingshook Biswas
Journal: Proc. Amer. Math. Soc. 142 (2014), 2013-2017
MSC (2010): Primary 37F50
Published electronically: March 11, 2014
MathSciNet review: 3182020
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Abstract: Let $ f(z) = e^{2\pi i \alpha }z + O(z^2), \alpha \in \mathbb{R}$, be a germ of a holomorphic diffeomorphism in $ \mathbb{C}$. For $ \alpha $ rational and $ f$ of infinite order, the space of conformal conjugacy classes of germs topologically conjugate to $ f$ is parametrized by the Ecalle-Voronin invariants (and in particular is infinite-dimensional). When $ \alpha $ is irrational and $ f$ is nonlinearizable it is not known whether $ f$ admits quasi-conformal deformations. We show that if $ f$ has a sequence of repelling periodic orbits converging to the fixed point, then $ f$ embeds into an infinite-dimensional family of quasi-conformally conjugate germs, no two of which are conformally conjugate.

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Kingshook Biswas
Affiliation: Department of Mathematics, Ramakrishna Mission Vivekananda University, Belur Math, Howrah 711202, India

Received by editor(s): September 13, 2011
Received by editor(s) in revised form: April 18, 2012, and May 27, 2012
Published electronically: March 11, 2014
Additional Notes: This research was partly supported by the Department of Science and Technology research project grant DyNo. 100/IFD/8347/2008-2009
Communicated by: Mario Bonk
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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