Univalent polynomials and Hubbard trees
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- by Kirill Lazebnik, Nikolai G. Makarov and Sabyasachi Mukherjee PDF
- Trans. Amer. Math. Soc. 374 (2021), 4839-4893 Request permission
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).References
Additional Information
- Kirill Lazebnik
- Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
- Address at time of publication: Department of Mathematics, University of Toronto, Toronto, Canada
- MR Author ID: 1006341
- ORCID: 0000-0001-8963-4410
- Email: kylazebnik@gmail.com
- Nikolai G. Makarov
- Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
- MR Author ID: 210350
- Email: makarov@caltech.edu
- Sabyasachi Mukherjee
- Affiliation: School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400005, India
- MR Author ID: 1098266
- ORCID: 0000-0002-6868-6761
- Email: sabya@math.tifr.res.in
- Received by editor(s): February 12, 2020
- Received by editor(s) in revised form: October 4, 2020
- Published electronically: April 28, 2021
- Additional Notes: 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.
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 4839-4893
- MSC (2020): Primary 30C10, 37F10
- DOI: https://doi.org/10.1090/tran/8387
- MathSciNet review: 4273178