Divergence and convergence of conjugacies in non-Archimedean dynamics

Abstract

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. 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. We find a lower (sometimes the best) bound of the size of the corresponding linearization disc. In the cases where we find the exact size of the linearization disc, we show, using the Weierstrass degree of the conjugacy, that f has an indifferent periodic point on the boundary. We also give a class of polynomials containing a monomial of degree prime to p, such that the conjugacy diverges. (2000): 32P05, 32H50, 37F50, 11R58 Mathematics Subject Classification

Linearization in ultrametric dynamics in fields of characteristic zero — Equal characteristic case

“…Similar estimates in the remaning cases, i.e. the cases in which K is either a p-adic field or a field of prime characteristic, were obtained in various papers on the p-adic case [5,18,35,42] later generalized in [28], and in [29,31] concerning the prime characteristic case. (2000): 32P05, 32H50, 37F50…”

“…One might therefore conjecture, as Herman [14], that for a locally compact, complete valued field of prime characteristics, the formal conjugacy 'usually' diverges, even for polynomials of one variable. Indeed, as shown in the papers [29,31] like in complex dynamics, the formal solution may diverge also in the one-dimensional case. On the other hand, in [29,31] it was also proven that the conjugacy may still converge due to considerable cancellation of small divisor terms; the same multipler λ may yield convergence for some f but not for others.…”

“…Indeed, as shown in the papers [29,31] like in complex dynamics, the formal solution may diverge also in the one-dimensional case. On the other hand, in [29,31] it was also proven that the conjugacy may still converge due to considerable cancellation of small divisor terms; the same multipler λ may yield convergence for some f but not for others. This brings about a problem of a combinatorial nature of seemingly great complexity and a complete description is yet to be found.…”

“…the maximal disc U , about the origin, such that the full conjugacy g • f • g −1 (x) = λx, holds for all x ∈ U . Estmates of linearization discs have appeard in several papers concerning the p-adic case [5,18,35,42] later generalized in [28], and in [29,31] concerning the prime characteristic case.…”

The size of quadratic p -adic linearization disks

“…In the multi-dimensional p-adic case, there exist multipliers λ such that the corresponding Siegel condition is violated and the conjugacy diverges [11,30]. In fields of positive characteristics, the convergence of the linearization series is far from trivial even in the one-dimensional case [17,19]. For results in fields of caracteristic zero-equal characteristic case, see [18].…”