On Siegel's linearization theorem for fields of prime characteristic

Abstract

In 1981, Herman andYoccoz (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).

Divergence and convergence of conjugacies in non-Archimedean dynamics

“…However, they also showed that the condition in Siegel's theorem is 'usually' not satisfied over fields of prime characteristic. Indeed, as proven in [21], there exist power series f such that the associated conjugacy function diverges. We prove that if the degrees of the monomials of a power series f are divisible by p, then f is analytically linearizable.…”

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We continue the study in [21] of the linearizability near an indifferent fixed point of a power series f , defined over a field of prime characteristic p. It is known since the work of Herman and Yoccoz [13] in 1981 that Siegel's linearization theorem [27] is true also for non-Archimedean fields. However, they also showed that the condition in Siegel's theorem is 'usually' not satisfied over fields of prime characteristic.

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On the chaotic behavior of a generalized logistic p-adic dynamical system

“…Then from the equality 2z 2 0 − az 0 + 6 + a 2 p = 2 3 (3z 0 − 1) 2 + 4 − a 3 (3z 0 − 1) + 1 3 3a 2 − a + 20 p (5. 32) we conclude that |2z 2 0 − az 0 + 6 + a 2 | p < 1 if and only if |3a 2 − a + 20| p < 1. By means of (5.25) and the last condition, we can formulate the following Corollary 5.11.…”

Generic parabolic points are isolated in positive characteristic

“…In contrast, for an ultrametric ground field of characteristic zero only the first alternative occurs: Every irrationally indifferent cycle is locally linearizable [HY83], and hence every point in the cycle is isolated as a periodic point. The case of an ultrametric field of positive characteristic is more subtle, since irrationally indifferent cycles are usually not locally linearizable, see for example [Lin04,Theorem 2.3] or [Lin10, Theorem 1.1]. Nevertheless, every irrationally indifferent periodic point is isolated as a periodic point [LRL16, Corollary 1.1].…”