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TOTALLY RAMIFIED RATIONAL MAPS

Cui, Weiwei LU and Hu, Jun (2022) In Conformal Geometry and Dynamics 26. p.208-234
Abstract

Totally ramified rational maps and regularly ramified rational maps are defined and studied in this paper. We first give a complete classification of regularly ramified rational maps and show that our definition of a regularly ramified rational map is equivalent to a much stronger definition of a map of this kind given by Milnor [Dynamics in one complex variable, Princeton University Press, Princeton, NJ, 2006]. Then we show that (1) any totally ramified rational map of degree d ≤ 6 must be regularly ramified; (2) for any integer d > 6, there exists a totally ramified rational map of degree d which is not regularly ramified. Furthermore, we count totally ramified rational maps up to degree 10. Finally, we present explicit formulas... (More)

Totally ramified rational maps and regularly ramified rational maps are defined and studied in this paper. We first give a complete classification of regularly ramified rational maps and show that our definition of a regularly ramified rational map is equivalent to a much stronger definition of a map of this kind given by Milnor [Dynamics in one complex variable, Princeton University Press, Princeton, NJ, 2006]. Then we show that (1) any totally ramified rational map of degree d ≤ 6 must be regularly ramified; (2) for any integer d > 6, there exists a totally ramified rational map of degree d which is not regularly ramified. Furthermore, we count totally ramified rational maps up to degree 10. Finally, we present explicit formulas for all totally but not regularly ramified rational maps of degree 7 or 8, up to pre and post-composition by Möbius transformations.

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author
and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
branched covering map, Ramified rational maps, Speiser graph
in
Conformal Geometry and Dynamics
volume
26
pages
27 pages
publisher
American Mathematical Society (AMS)
external identifiers
  • scopus:85149671210
ISSN
1088-4173
DOI
10.1090/ecgd/377
language
English
LU publication?
yes
id
0deb43d4-c963-4907-94ec-b27bddd85c77
date added to LUP
2023-04-04 11:20:20
date last changed
2025-04-04 14:32:39
@article{0deb43d4-c963-4907-94ec-b27bddd85c77,
  abstract     = {{<p>Totally ramified rational maps and regularly ramified rational maps are defined and studied in this paper. We first give a complete classification of regularly ramified rational maps and show that our definition of a regularly ramified rational map is equivalent to a much stronger definition of a map of this kind given by Milnor [Dynamics in one complex variable, Princeton University Press, Princeton, NJ, 2006]. Then we show that (1) any totally ramified rational map of degree d ≤ 6 must be regularly ramified; (2) for any integer d &gt; 6, there exists a totally ramified rational map of degree d which is not regularly ramified. Furthermore, we count totally ramified rational maps up to degree 10. Finally, we present explicit formulas for all totally but not regularly ramified rational maps of degree 7 or 8, up to pre and post-composition by Möbius transformations.</p>}},
  author       = {{Cui, Weiwei and Hu, Jun}},
  issn         = {{1088-4173}},
  keywords     = {{branched covering map; Ramified rational maps; Speiser graph}},
  language     = {{eng}},
  pages        = {{208--234}},
  publisher    = {{American Mathematical Society (AMS)}},
  series       = {{Conformal Geometry and Dynamics}},
  title        = {{TOTALLY RAMIFIED RATIONAL MAPS}},
  url          = {{http://dx.doi.org/10.1090/ecgd/377}},
  doi          = {{10.1090/ecgd/377}},
  volume       = {{26}},
  year         = {{2022}},
}