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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Quasi-conformal deformations of nonlinearizable germs
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by Kingshook Biswas PDF
Proc. Amer. Math. Soc. 142 (2014), 2013-2017 Request permission

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.
References
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Additional Information
  • Kingshook Biswas
  • Affiliation: Department of Mathematics, Ramakrishna Mission Vivekananda University, Belur Math, Howrah 711202, India
  • Email: kingshook@rkmvu.ac.in
  • 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
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 142 (2014), 2013-2017
  • MSC (2010): Primary 37F50
  • DOI: https://doi.org/10.1090/S0002-9939-2014-11896-X
  • MathSciNet review: 3182020