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.

When I started the bounty there were already two fairly good answers to this question. One, by Agol, points to the theory of post-critically finite maps. The other, by David Speyer, suggests an approach to count the number of such maps if there are any. Any other ideas ?

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.

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.

When I started the bounty there were already two fairly good answers to this question. One, by Agol, points to the theory of post-critically finite maps. The other, by David Speyer, suggests an approach to count the number of such maps if there are any. Any other ideas ?

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.

What can be said about rational self-maps of P(1) $\mathbb P^1$ 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.

What can be said about rational self-maps of 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 ( Corollary 1 ).

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.