Shapes, fingerprints and rational lemniscates
Author:
Malik Younsi
Journal:
Proc. Amer. Math. Soc. 144 (2016), 1087-1093
MSC (2010):
Primary 37E10, 30C20; Secondary 30F10
DOI:
https://doi.org/10.1090/proc12751
Published electronically:
June 30, 2015
MathSciNet review:
3447662
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Abstract | References | Similar Articles | Additional Information
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 is given by the
-th root of a Blaschke product of degree
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
Email:
malik.younsi@gmail.com
DOI:
https://doi.org/10.1090/proc12751
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