TITLE: Rational maps with all critical points fixed
QUESTION [13 upvotes]: What can be said about rational self-maps of $\mathbb P^1$
for which all critical points are also fixed points ?
If all but one of the fixed points are critical, there is
a characterization in http://arxiv.org/abs/math/0411604v1 
( see Corollary 1 and the discussion just after the statement ).
Still assuming that all critical points are fixed: Is it possible to bound the degree of the rational map if all but two of the fixed points are critical ?
I think that the answer is probably no, but I would really
love to hear the contrary.

Motivation. The question is motivated by a rather specific problem I like to think about from time to time. It concerns the 
classification of some special arrangements of lines on the projective plane. More specifically, I would like to classify arrangements
of $3d$ lines(or rather hyperplanes through the origin of $\mathbb C^3$) invariant by degree $d$ homogeneous polynomial vector fields on $\mathbb C^3$. Given one arrangement like that one can produce a degree $d$ rational map having all its critical points fixed. 

Update. (08/28/2013) The paper On the classification of critically fixed rational maps by Cordwell, Gilbertson, Nuechterlein, Pilgrim, and Pinella classifies rational maps for which all critical points are fixed. In particular, for every degree $d\ge 3$ there exists a rational  such that all but two of the fixed points are critical.

REPLY [6 votes]: Theorem: for each n, there are up to Mobius conjugacy a finite, nonzero, number of rational maps of degree n such that all of the critical points are fixed.
Proof: Nonempty is clear: z^n.  For finiteness: the degree bounds the number of critical points. Given a bound on the degree and the size of the postcritical set, with the usual set of flexible Lattes exceptions (which have non-fixed critical points), the set of such rational maps up to Mobius conjugacy is finite; this is a consequence of Thurston's Rigidity Theorem [Douady, Adrien; Hubbard, John H. A proof of Thurston's topological characterization of rational functions. Acta Math. 171 (1993), no. 2, 263–297.]; cf. also section 3 of: [E. Brezin, R. Byrne, J. Levy, K. Pilgrim, and K. Plummer). Conf. Geom. & Dynam. Sys. (electronic) 4(2000), 35-74. ]