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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 for all totally but not regularly ramified rational maps of degree 7 or 8, up to pre-and post-composition by Möbius transformations.