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Shapes, fingerprints and rational lemniscates

Author: Malik Younsi
Journal: Proc. Amer. Math. Soc. 144 (2016), 1087-1093
MSC (2010): Primary 37E10, 30C20; Secondary 30F10
Published electronically: June 30, 2015
MathSciNet review: 3447662
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Abstract: It has been known for a long time that any smooth Jordan curve in the plane can be represented by its so-called fingerprint, an orientation preserving smooth diffeomorphism of the unit circle onto itself. In this paper, we give a new, simple proof of a theorem of Ebenfelt, Khavinson and Shapiro stating that the fingerprint of a polynomial lemniscate of degree $ n$ is given by the $ n$-th root of a Blaschke product of degree $ n$ and that, conversely, any smooth diffeomorphism induced by such a map is the fingerprint of a polynomial lemniscate of the same degree. The proof is easily generalized to the case of rational lemniscates, thus solving a problem raised by the previously mentioned authors.

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Additional Information

Malik Younsi
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651

Keywords: Lemniscates, fingerprints, rational maps, conformal representation.
Received by editor(s): June 13, 2014
Received by editor(s) in revised form: February 6, 2015
Published electronically: June 30, 2015
Additional Notes: This research was supported by NSERC
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2015 American Mathematical Society

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