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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
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