Shapes, fingerprints and rational lemniscates

Younsi, Malik

doi:10.1090/proc12751

Shapes, fingerprints and rational lemniscates
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Proc. Amer. Math. Soc. 144 (2016), 1087-1093 Request permission

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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