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Non-existence of prescribable conformally equivariant dilatation in space


Authors: Malinee Chaiya and Aimo Hinkkanen
Journal: Proc. Amer. Math. Soc. 141 (2013), 3985-3995
MSC (2010): Primary 30C65; Secondary 30C62
DOI: https://doi.org/10.1090/S0002-9939-2013-11692-8
Published electronically: August 1, 2013
MathSciNet review: 3091789
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Abstract: In this paper, we study the prescribable conformally equivariant dilatations for orientation preserving quasiconformal homeomorphisms. The complex dilatation is a prescribable conformally equivariant dilatation in $ \mathbb{R}^{2}$. A Schottky set is a subset of the unit sphere $ \mathbb{S}^n$ whose complement is the union of at least three disjoint open balls. By using the result of Bonk, Kleiner, and Merenkov that there are rigid Schottky sets of positive measure in each dimension at least $ 3$, we prove that it is not possible to have a prescribable conformally equivariant dilatation in $ \mathbb{R}^{n}$, where $ n \geq 3$.


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Additional Information

Malinee Chaiya
Affiliation: Department of Mathematics, Faculty of Science, Silpakorn University, Nakorn Pathom 73000, Thailand
Email: malinee.c@su.ac.th

Aimo Hinkkanen
Affiliation: Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 W. Green Street, Urbana, Illinois 61801
Email: aimo@math.uiuc.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11692-8
Received by editor(s): March 21, 2011
Received by editor(s) in revised form: October 7, 2011, and January 30, 2012
Published electronically: August 1, 2013
Additional Notes: This material is based upon work supported by the National Science Foundation under Grants No. 0758226 and 1068857.
Communicated by: Mario Bonk
Article copyright: © Copyright 2013 American Mathematical Society