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.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:vxu-1713 |
| Date | January 2007 |
| Creators | Lindahl, Karl-Olof |
| Publisher | Växjö universitet, Matematiska och systemtekniska institutionen, Växjö : Växjö University Press |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Doctoral thesis, monograph, info:eu-repo/semantics/doctoralThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | Acta Wexionensia, 1404-4307 ; 1404-4307 |
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