## The field of definition for dynamical systems on $$\mathbb{P}^ 1$$.(English)Zbl 0849.11090

The author investigates a structural question in algebraic dynamics. Take a rational mapping $$\varphi$$ over a field $$K$$. Conjugacy by $$PGL_2$$ yields the equivalence class $$[ \varphi ]$$. Conjugacy does not change the dynamics, but it may change the field of definition, giving a collection of fields for each $$[ \varphi ]$$. They are all extensions of the field of moduli $$L$$, which is the fixed field of the subgroup of $$\text{Gal} (\overline {K}/ K)$$ constituted by the automorphisms conjugating $$\varphi$$ to a mapping of the same class. So if $$L$$ happens to be a field of definition for $$[ \varphi ]$$, then it is the smallest one.
The author asks when is $$L$$ a field of definition for $$[ \varphi ]$$, and shows that this is the case if $$\varphi$$ has even degree, or is conjugated to a polynomial. He also finds conditions under which the field of moduli of a class is not a field of definition for any of its elements.
A clear style and good organization make pleasant reading and help the uninitiated. I have just one complaint. The author – an algebraist – defines a dynamical system to be a conjugacy class of rational maps under $$PGL_2$$. The reader gets the impression that no other construct qualifies as a dynamical system. As a dynamicist, I shall then retaliate and define the rings to be those subsets of $$\mathbb{Z}$$ constituted by the multiples of some odd power of 19.
Reviewer: F.Vivaldi (London)

### MSC:

 11R99 Algebraic number theory: global fields 37-XX Dynamical systems and ergodic theory 14E05 Rational and birational maps

### Keywords:

dynamical systems; field of definition; field of moduli
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### References:

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