# Licentiate seminar in mathematics: Jonas Nordqvist

**Title:** Ramification numbers and periodic points in arithmetic dynamical systems**Subject:** Mathematics**Faculty:** Faculty of Technology**Date:** Thursday 15 February 2018 at 1.00 pm**Location:** Room D1136, building D, Växjö**External reviewer:** Professor Johan Öinert, Blekinge Institute of Technology (BTH), Sweden**Chairman: **Professor Joachim Toft, Department of Mathematics, Linnaeus University**Supervisor:** Associate Professor Karl-Olof Lindahl, Department of Mathematics, Linnaeus University**Examiner:** Professor Andrei Khrennikov, Department of Mathematics, Linnaeus University

## Abstract

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* + ... + *p ^{n}*). Moreover, we also obtain a lower bound of the absolute value of the periodic points in the open unit disk of such series.