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

Abstract

Let K be a complete ultrametric field of charactersitic zero whose corresponding residue field k is also of charactersitic zero. We give lower and upper bounds for the size of linearization disks for power series over K near an indifferent fixed point. These estimates are maximal in the sense that there exist exemples where these estimates give the exact size of the corresponding linearization disc. 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 Mathematics Subject Classification

Estimates of linearization discs in $p$-adic dynamics with application to ergodicity

“…In dimension one, the condition (2) is always satisfied for non-resonant multipliers in fields of characteristic zero, i.e. the p-adic case studied in this paper, and the equal characteristic case of studied in [47]. This is not always true in fields of prime characteristic as shown in [44,46].…”

“…Further results on the properties of the dynamics on p-adic linearization discs are provided in [4,53]. Estimates for linearization discs in prime characteristic were obtained in [44,46], and for fields of charactersitic zero in the equal charactersitic case [47]. See [44], for further comments on the non-Archimedean problem of linearization and its relation to the complex field case.…”

“…In particular, in unramified extensions of Qp, the linearization disc is maximal if the multiplier map has a maximal cycle on the unit sphere. Estimates of linearization discs in the remaining types of non-Archimedean fields of dimension one were obtained in [44,46,47].Moreover, it is shown that, for any complete non-Archimedean field, transitivity is preserved under analytic conjugation. Using results by Oxtoby [52], we prove that transitivity, and hence minimality, is equivalent the unique ergodicity on compact subsets of a linearization disc.…”

“…In particular, in unramified extensions of Qp, the linearization disc is maximal if the multiplier map has a maximal cycle on the unit sphere. Estimates of linearization discs in the remaining types of non-Archimedean fields of dimension one were obtained in [44,46,47].…”

The size of quadratic p -adic linearization disks

“…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]. In the hyperbolic case |λ| = 1, 0, the linearization disk will in general be the maximal disk of injectivity [20].…”