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Residue fixed point index and wildly ramified power series
Linnaeus University, Faculty of Technology, Department of Mathematics.ORCID iD: 0000-0002-0510-6782
2020 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis concerns discrete dynamical systems. These are systems where the dynamics is modeled by iterated functions. There are several applications of discrete dynamical system e.g. in biology, pseudo random number generation and statistical mechanics. In this thesis we are interested in discrete dynamical systems described by iterations of a power series f fixing the origin, where it is tangent to the identity. In particular, the coefficients of f are given in a field of positive characteristic p. We are interested in the so-called lower ramification numbers of such series. The lower ramification numbers encodes the multiplicity of the origin as a fixed point of f under p-power iterates. In particular this thesis contains four papers all related to the topic of lower ramification numbers of such power series.

In Paper I we consider so-called 2-ramified power series and give a characterization of such in terms of its first significant terms. This is further extended in Paper II, where we geometrically locate the periodic points of 2-ramified power series in the open unit disk. In doing, so we provide a self-contained proof of the main result of the first paper.

In Paper III, we consider power series with a fixed point at the origin of small multiplicity, i.e. the multiplicity of the fixed point is less than that of the characteristic of the ground field. We provide a characterization of all such power series having the smallest possible lower ramification numbers, in terms of its first significant terms, and in terms of the nonvanishing of the so-called iterative residue. In doing so, we also provide a formula for the residue fixed point index for the case of a multiple fixed point. We further extend the results of Paper II by locating geometrically the periodic points in the open unit disk of convergent power series with small multiplicity.

In Paper IV we consider power series of large multiplicity, and introduce an invariant in positive characteristic closely related to the residue fixed point index. We provide a characterization of these power series having the smallest possible lower ramification numbers in terms of the nonvanishing of this invariant. As a by-product we obtain results about the dimension of the moduli space of formal classification of wildly ramified power series.

Abstract [sv]

Denna avhandling behandlar diskreta dynamiska system. Dessa är system där dynamiken modelleras med en itererad funktion. Tillämpningarna för diskreta dynamiska systems återfinns bland annat i biologi, pseudo-slumptalsgenerering och statistisk mekanik för att nämna några exempel. I denna avhandling är vi intresserade av dynamiska system som beskrivs av iterationer av en potensserie f med en fixpunkt i origo, där den tangerar identitetsavbildningen. Särskilt är vi intresserade av potensserier vars koefficienter kommer från en kropp med positiv karaktäristik p. Vi är intresserade av de så kallade underförgreningstalen till f. Underförgreningstalen till en potensserie f kan kortfattat beskrivas som multipliciteten av origo betraktad som en fixpunkt till iterationer av f. Denna avhandling innehåller fyra artiklar som behandlar detta ämne.

I artikel I studerar vi så kallade 2-förgrenade potensserier och karaktäriserar dessa i termer av dess första signifikanta termer. Detta utvecklas sedan i artikel II, där vi beskriver absolutbeloppet av de periodiska punkterna för 2-förgrenade potensserier. Detta medför också att vi erhåller ett fristående bevis för huvudresultatet i artikel I.

 

I artikel III, studerar vi potensserier med en fixpunkt i origo med liten multiplicitet, d.v.s. potensserier där multipliciteten för origo som en fixpunkt är mindre än karaktäristiken för koefficientkroppen. Vi erhåller en karaktärisering av dessa i termer av dess första signifikanta termer, samt även formulerat som ickeförsvinnandet av den så kallade iterativa residyn. Vi erhåller även en formel för att beräkna det holomorfa indexet för en fixpunkt i fallet då multiplikatorn av fixpunkten är 1. Vidare utvecklar vi resultaten från artikel II genom att erhålla resultat om de periodiska punkternas geometriska placering i den öppna enhetsskivan för konvergenta potensserier med en fixpunkt med liten multiplicitet.

I artikel IV studerar vi potensserier där origo är en fixpunkt med multiplicitet som är större än karaktäristiken i koefficientkroppen. Vi introducerar även en invariant i positiv karaktäristik nära besläktad med det holomorfa indexet. Vi erhåller en karaktärisering av sådana potensserier när dessa har de minsta möjliga underförgreningstalen, i termer av ickeförsvinnandet av denna invariant. Vi erhåller också resultat gällande antalet parametrar som behövs för att klassificera dessa potensserier under formella koordinatbyten.

Place, publisher, year, edition, pages
Växjö: Linnaeus University Press, 2020. , p. 136
Series
Linnaeus University Dissertations
Keywords [en]
wildly ramified power series, residue fixed point index, formal invariant, iterative residue, ramification numbers, non-archimedean, discrete dynamical systems, arithmetic dynamics, periodic points, Nottingham group
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
URN: urn:nbn:se:lnu:diva-90867ISBN: 978-91-89081-30-7 (print)ISBN: 978-91-89081-31-4 (electronic)OAI: oai:DiVA.org:lnu-90867DiVA, id: diva2:1384986
Public defence
2020-02-06, Weber, Hus K, Växjö, 14:00 (English)
Opponent
Supervisors
Available from: 2020-01-13 Created: 2020-01-13 Last updated: 2020-01-13Bibliographically approved
List of papers
1. Characterization of 2-ramified power series
Open this publication in new window or tab >>Characterization of 2-ramified power series
2017 (English)In: Journal of Number Theory, ISSN 0022-314X, E-ISSN 1096-1658, Vol. 174, p. 258-273Article in journal (Refereed) Published
Abstract [en]

In this paper we study lower ramification numbers of power series tangent to the identity that are defined over fields of positive characteristics p. Let g be such a series, then g has a fixed point at the origin and the corresponding lower ramification numbers of g are then, up to a constant, the degree of the first non-linear term of p-power iterates of g. The result is a complete characterization of power series g having ramification numbers of the form 2 ( 1 + p + 
 + p n ) . Furthermore, in proving said characterization we explicitly compute the first significant terms of g at its pth iterate.

Place, publisher, year, edition, pages
Elsevier, 2017
Keywords
Lower ramification numbers, iterations of power series, difference equations, arithmetic dynamics
National Category
Other Mathematics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-58627 (URN)10.1016/j.jnt.2016.10.005 (DOI)000392902700016 ()2-s2.0-85006507544 (Scopus ID)
Note

Correction published in: Nordqvist, Jonas. 2017. Corrigendum to “Characterization of 2-ramified power series” [J. Number Theory 174 (2017) 258–273], Journal of Number Theory, 178: 208.

Available from: 2016-12-02 Created: 2016-12-02 Last updated: 2020-01-13Bibliographically approved
2. Geometric location of periodic points of 2-ramified power series
Open this publication in new window or tab >>Geometric location of periodic points of 2-ramified power series
2018 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 465, no 2, p. 762-794Article in journal (Refereed) Published
Abstract [en]

In this paper we study the geometric location of periodic points of power series defined over fields of prime characteristic p. More specifically, we find a lower bound for the absolute value of all periodic points in the open unit disk of minimal period pn of 2-ramified power series. We prove that this bound is optimal for a large class of power series. Our main technical result is a computation of the first significant terms of the pnth iterate of 2-ramified power series. As a by-product we obtain a self-contained proof of the characterization of 2-ramified power series.

Place, publisher, year, edition, pages
Elsevier, 2018
National Category
Geometry
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-64440 (URN)10.1016/j.jmaa.2018.05.009 (DOI)
Available from: 2017-05-29 Created: 2017-05-29 Last updated: 2020-01-13Bibliographically approved

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