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On the linearization of non-Archimedean holomorphic functions near an indifferent fixed point

We consider the problem of local linearization of power series defined over complete valued fields. The complex field case has been studied since the end of the nineteenth century, and renders a delicate number theoretical problem of small divisors related to diophantine approximation. Since a work of Herman and Yoccoz in 1981, there has been an increasing interest in generalizations to other valued fields like p-adic fields and various function fields. We present some new results in this domain of research. In particular, for fields of prime characteristic, the problem leads to a combinatorial problem of seemingly great complexity, albeit of another nature than in the complex field case. In cases for which linearization is possible, we estimate the size of linearization discs and prove existence of periodic points on the boundary. We also prove that transitivity and ergodicity is preserved under the linearization. In particular, transitivity and ergodicity on a sphere inside a non-Archimedean linearization disc is possible only for fields of p-adic numbers.

Identiferoai:union.ndltd.org:UPSALLA1/oai:DiVA.org:vxu-1713
Date January 2007
CreatorsLindahl, Karl-Olof
PublisherVäxjö universitet, Matematiska och systemtekniska institutionen, Växjö : Växjö University Press
Source SetsDiVA Archive at Upsalla University
LanguageEnglish
Detected LanguageEnglish
TypeDoctoral thesis, monograph, info:eu-repo/semantics/doctoralThesis, text
Formatapplication/pdf
Rightsinfo:eu-repo/semantics/openAccess
RelationActa Wexionensia, 1404-4307 ; 1404-4307

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