# Public defence in mathematics: Jonas Nordqvist

**Title**: Residue fixed point index and wildly ramified power series**Subject**: Mathematics**Faculty**: Faculty of Technology**Date**: Thursday 6 February 2020 at 2.00 pm**Place**: Room Weber, building K, Växjö**External reviewer**: Professor Robert L Benedetto, Amherst College, USA**Examining committee**: Professor Johan Öinert, Blekinge tekniska högskola, Sweden

Associate professor Tomas Persson, Lund University, Sweden

Professor Elizabeth Wulcan, Chalmers University of Technology, Sweden**Chairperson**: Professor Joachim Toft, Department of Mathematics, Linnaeus University**Supervisor**: Associate professor Karl-Olof Lindahl, Department of Mathematics, Linnaeus University**Assistant supervisor:** Dr Per-Anders Svensson, Department of Mathematics, Linnaeus University**Examiner**: Professor Andrei Khrennikov, Department of Mathematics, Linnaeus University

**Spikning**: Friday 10 January 2020 at 11.00 am at the University library in 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.