Kreishomöomorphismen - Aspekte eines dynamischen Systems.

In this work we will provide a comprehensive overview of a several aspects of circle maps. Firstly, the rotation number is introduced, a central property of circle maps that will later be used to make a complete classification of orbit types. There are six types of orbit types: Three each for the cases irrational and rational rotation number. Additionally, we will show with Denjoy’s theorem that sharper requirements for smoothness reduces the types of orbits in the irrational case. In the subsequent chapter, the space of circle maps will be studied as a topological object. We will have a look at equivalence classes with respect to approximated conjugation. After that, we will look at ergodic theory for circle maps. Weyl’s theorem will be shown and used to prove that the sequence \(2n\) obeys Benford’s Law. In the last chapter we will explore the relation of rotation number and the parameters \(a\) and \(\epsilon\) of the family of maps \(x \mapsto x+a+ \frac{\epsilon}{2\pi} \cos(x)\). The phenomenon of mode locking can be observed for certain pairs of \(a\) and (\epsilon\) and will be studied.