The Galois Theory of Periodic Points of Polynomial Maps

0. Introduction There is a growing body of evidence that problems associated with iterated maps can yield interesting insights in number theory and algebra. There are the papers of Narkiewicz [19-22], Lewis [14], Liardet [15], and more recently, Silverman [27], dealing with the diophantine aspects of iterated maps, as well as the papers of Odoni [24,25], Stoll [30], Vivaldi and Hatjispyros [34] (also see [33]), and Morton [16,17] (also see [2]), which deal with the Galois-theoretic aspects of iterated maps. In [24] and [25] Odoni is concerned with the Galois groups of pure iterates of polynomials, while in [34] Vivaldi and Hatjispyros deal with Galois groups of periodic points of polynomial maps (especially quadratic maps). The overview of Lagarias [12] contains many other interesting connections between dynamical systems and number theory. This paper is also concerned with the Galois groups of periodic points, and in particular, with the question of whether or not a given polynomial map acts as an automorphism on the field generated by its periodic points. In §§ 2 and 3 we establish some basic results about the polynomials whose roots are the periodic points of primitive (or minimal) period n of a polynomial map o(x) defined over a field K. In §§ 4 and 5 we use these results to study the Galois group of the field !„ generated over K by the periodic points of o of primitive period n, and in §§ 5 and 6 we show how these Galois groups can be computed with the help of a set of natural resolvents. (See also [34].) In § 7 we derive conditions under which the map o is an automorphism on the field 22. We finish the paper by proving that the generic polynomial o is an automorphism of 2W, in a precise sense described more explicitly below. In [2] the results in §§ 2 and 3 are used to study the dynamics of polynomial maps over finite fields, and in [18] the results of § 2 are generalized and used to exhibit units in the algebraic number fields generated by periodic points of polynomials (and more general maps) over Q. After a preliminary version of this paper had been written, we learned of the results of [34], which have some overlap with the results in §§4 and 5. We are grateful to F. Vivaldi and J. Silverman for their remarks concerning an earlier version of this paper, and to J. Shallit for making us aware of several references. We are also grateful to the referee for numerous detailed comments on the exposition.