On Siegel's linearization theorem for fields of prime characteristic

Abstract

In 1981, Herman and Yoccoz (1983 Generalizations of some theorems of small divisors to non Archimedean fields Geometric Dynamics (Lecture Notes in Mathematics) ed J Palis Jr, pp 408–47 (Berlin: Springer) Proc. Rio de Janeiro, 1981) proved that Siegel's linearization theorem (Siegel C L 1942 Ann. Math. 43 607–12) is true also for non-Archimedean fields. However, the condition in Siegel's theorem is usually not satisfied over fields of prime characteristic. We consider the following open problem from non-Archimedean dynamics. Given an analytic function f defined over a complete, non-trivial valued field of characteristic p > 0, does there exist a convergent power series solution to the Schröder functional equation (2) that conjugates f to its linear part near an indifferent fixed point? We will give both positive and negative answers to this question, one of the problems being the presence of small divisors. When small divisors are present this brings about a problem of a combinatorial nature, where the convergence of the conjugacy is determined in terms of the characteristic of the state space and the powers of the monomials of f, rather than in terms of the diophantine properties of the multiplier, as in the complex case. In the case that small divisors are present, we show that quadratic polynomials are analytically linearizable if p = 2. We find an explicit formula for the coefficients of the conjugacy, and applying a result of Benedetto (2003 Am. J. Math. 125 581–622), we find the exact size of the corresponding Siegel disc and show that there is an indifferent periodic point on the boundary. In the case p > 2 we give a sufficient condition for divergence of the conjugacy for quadratic maps as well as for a certain class of power series containing a quadratic term (corollary 2.1).