Univalent polynomials and Hubbard trees

We study rational functions f of degree d+1 such that f is univalent in the exterior unit disc, and the image of the unit circle under f has the maximal number of cusps (d+1) and double points (d-2). We introduce a bi-angled tree associated to any such f. It is proven that any bi-angled tree is real...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2021-07, Vol.374 (7), p.4839-4893
Main Authors: Lazebnik, Kirill, Makarov, Nikolai G., Mukherjee, Sabyasachi
Format: Article
Language:English
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Summary:We study rational functions f of degree d+1 such that f is univalent in the exterior unit disc, and the image of the unit circle under f has the maximal number of cusps (d+1) and double points (d-2). We introduce a bi-angled tree associated to any such f. It is proven that any bi-angled tree is realizable by such an f, and moreover, f is essentially uniquely determined by its associated bi-angled tree. This combinatorial classification is used to show that such f are in natural 1:1 correspondence with anti-holomorphic polynomials of degree d with d-1 distinct, fixed critical points (classified by their Hubbard trees).
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/8387