Universität Hamburg - Fachbereich Mathematik - AG Angewandte Mathematik

Bodo Werner: Research Project Invariant Curves, Circle Maps and their Numerical Analysis.

Invariant curves in our understanding are smooth closed curves, invariant under a diffeomorphism. The analysis is based on the concept of treating a closed curve as a graph of a function. This allows the representation of an invariant curve as a fixed point of the graph transform and to formulate conditions which imply normal hyperbolicity. The PhD-thesis of Nils Nicolaisen on this subject from January 1998 is available. Additionally we are interested in orientation prserving circle maps given for instance by the restriction of the diffeomorphism to an invariant curve. Here we aim on invariant (ergodic) measures, see

Discretization of circle maps (with N. Nicolaisen), Z. angew. Math. Phys. 49 (1998) 869-895 Here the Frobenius-Perron operator is discretized by some finite partition of the circle which leads to a transition matrix representing a finite Markov chain. The aim is to obtain characteristics of the circle map by its transition matrix.

In the paper

The spectrum of the Frobenius-Perron operator and its discretization for circle diffeomorphisms (with N. Lehmkuhl and W. Hotzel), Hamburger Beiträge zur Angewandten Mathematik, Reihe A, Preprint 160, January 2001 File: hbam2001a160.ps.gz

we ask what information about the dynamic can be obtained from the spectrum.

Two doctorate students are working on this project, Nicole Lehmkuhl and Wernt Hotzel.