Totally ramified rational maps

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
 
 d\leq 6
 

 must be regularly ramified; (2) for any integer 

 
 
 d
 >
 6
 
 d>6
 

, there exists a totally ramified rational map of degree 

 
 d
 d
 

 which is not regularly ramified. Furthermore, we count totally ramified rational maps up to degree 

 
 10
 10
 

. Finally, we present explicit formulas for all totally but not regularly ramified rational maps of degree 

 
 7
 7
 

 or 

 
 8
 8
 

, up to pre- and post-composition by Möbius transformations.