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Ramification numbers and periodic points in arithmetic dynamical systems
Linnaeus University, Faculty of Technology, Department of Mathematics.ORCID iD: 0000-0002-0510-6782
2018 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

The field of discrete dynamical systems is a rich and active field of research within mathematics, with applications ranging from biology to computer science, finance, engineering and various others. In this thesis properties of certain discrete dynamical systems are studied together with number theoretic properties of the functions defining these systems. The dynamical systems studied in this thesis are defined by iteration of power series g with a fixed point at the origin, tangent to the identity, and defined over fields of prime characteristic p. We are interested in the geometric location of the periodic points in the open unit disk. Recent results have shown that there is a connection between the lower ramification numbers of g and the geometric location of the periodic points in the open unit disk. The lower ramification numbers of g can be described as the multiplicity of zero as a fixed point of p-power iterates of g.

Part of this thesis concerns characterizing power series having certain sequences of ramification numbers. The other part concerns utilizing these results in order to describe the geometric location of the periodic points in terms of their distance to the origin. More precisely, we characterize all 2-ramified power series, i.e. power series having ramification numbers of the form 2(1 + p + … + pn). Moreover, we also obtain a lower bound of the absolute value of the periodic points in the open unit disk of such series.

Place, publisher, year, edition, pages
Växjö: Linnaeus University Press, 2018. , p. 74
Series
Lnu Licentiate ; 10
Keywords [en]
ramification numbers, local fields, arithmetic dynamics, periodic points, Nottingham group
National Category
Mathematics
Research subject
Natural Science, Mathematics
Identifiers
URN: urn:nbn:se:lnu:diva-69926ISBN: 978-91-88761-28-6 (print)ISBN: 978-91-88761-29-3 (electronic)OAI: oai:DiVA.org:lnu-69926DiVA, id: diva2:1175046
Presentation
2018-02-15, D1136, Växjö, 13:00 (English)
Opponent
Supervisors
Available from: 2018-01-17 Created: 2018-01-17 Last updated: 2018-01-17Bibliographically 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.

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 ()
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: 2018-08-30Bibliographically 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.

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: 2018-09-10Bibliographically approved

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