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Dynamical Belyi maps and arboreal Galois groups. (English) Zbl 1470.11178

Summary: We consider a large class of so-called dynamical Belyi maps and study the Galois groups of iterates of such maps. From the combinatorial invariants of the maps, we construct a useful presentation of the geometric Galois groups as subgroups of automorphism groups of regular trees, in terms of iterated wreath products. Using results on the reduction of dynamical Belyi maps modulo certain primes, we obtain results on the corresponding arithmetic Galois groups of iterates. These lead to results on the behavior of the arithmetic Galois groups under specialization, with applications to dynamical sequences.

MSC:

11G32 Arithmetic aspects of dessins d’enfants, Belyĭ theory
12F10 Separable extensions, Galois theory
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37P15 Dynamical systems over global ground fields
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References:

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