Abstract
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.
Highlights
Let f : P1K → P1K be a rational map defined over a number field K
The Galois theory of the iterates f n = f ◦ · · · ◦ f : P1K → P1K was first studied in the work of Odoni [13], and has applications both in number theory and in arithmetic dynamics
Specializing the iterates f n at a K -rational place a ∈ P1K, we obtain a tower of number fields (Kn,a)n≥1
Summary
Let f : P1K → P1K be a rational map defined over a number field K. The goal of this paper is to study the Galois group G∞,a and the primes ramifying in the corresponding tower of number fields for normalized Belyi maps. In analyzing the properties of the specialized Galois groups Gn,a (Theorem 3.2.2) our full understanding of the ramification structure of iterates of dynamical Belyi maps plays a key role. Another important ingredient is that the reduction behavior of a normalized Belyi map f , like its monodromy, can be completely expressed in terms of its combinatorial type, often yielding explicit and easy to apply criteria for good and bad reduction. This is applied to derive consequences for prime divisors of dynamical sequences in Corollary 4.2.1
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