Disputation i matematik: Jonas Nordqvist

Titel: Residue fixed point index and wildly ramified power series
Ämne: Matematik
Fakultet: Fakulteten för teknik
Datum: Torsdagen den 6 februari 2020 kl 14.00
Plats: Sal Weber, hus K, Växjö
Opponent: Professor Robert L Benedetto, Amherst College, USA
Betygsnämnd: Professor Johan Öinert, Blekinge tekniska högskola
Docent Tomas Persson, Lunds universitet
Biträdande professor Elizabeth Wulcan, Chalmers tekniska högskola
Ordförande: Professor Joachim Toft, institutionen för matematik, Linnéuniversitetet
Handledare: Docent Karl-Olof Lindahl, institutionen för matematik, Linnéuniversitetet
Biträdande handledare: Dr Per-Anders Svensson, institutionen för matematik, Linnéuniversitetet
Examinator: Professor Andrei Khrennikov, institutionen för matematik, Linnéuniversitetet
Spikning: Fredagen den 10 januari 2020 kl 11.00 på Universitetsbiblioteket i Växjö

Abstract

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.