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Univalent polynomials and Hubbard trees

Lazebnik, Kirill and Makarov, Nikolai G. and Mukherjee, Sabyasachi (2021) Univalent polynomials and Hubbard trees. Transactions of the American Mathematical Society, 374 (7). pp. 4839-4893. ISSN 0002-9947. doi:10.1090/tran/8387. https://resolver.caltech.edu/CaltechAUTHORS:20210715-154348375

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Abstract

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).


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1090/tran/8387DOIArticle
https://arxiv.org/abs/1908.05813arXivDiscussion Paper
ORCID:
AuthorORCID
Lazebnik, Kirill0000-0001-8963-4410
Mukherjee, Sabyasachi0000-0002-6868-6761
Additional Information:© 2021 American Mathematical Society. Received by the editors February 12, 2020, and, in revised form, October 4, 2020. Article electronically published on April 28, 2021. The third author was supported by the Institute for Mathematical Sciences at Stony Brook University, an endowment from Infosys Foundation and SERB research grant SRG/2020/000018 during parts of the work on this project. He also thanks Caltech for their support towards the project. The authors thank the anonymous referee for numerous valuable suggestions.
Funders:
Funding AgencyGrant Number
Stony Brook UniversityUNSPECIFIED
Infosys FoundationUNSPECIFIED
Science and Engineering Research Board (SERB)SRG/2020/000018
CaltechUNSPECIFIED
Issue or Number:7
Classification Code:2020 Mathematics Subject Classification: Primary: 30C10, 37F10
DOI:10.1090/tran/8387
Record Number:CaltechAUTHORS:20210715-154348375
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210715-154348375
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:109846
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:15 Jul 2021 16:55
Last Modified:15 Jul 2021 16:55

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