In this bachelor thesis we consider the dynamics of circle homeomorphisms. We define the circle as the set of equivalence classes and study homeomorphisms f that map from the circle to the circle. We look at a simple example of a circle homeomorphism: a rigid rotation, and through the study of orbits classify its dynamics. We define the lift of a homeomorphism as a function that lifts the circle maps to the real line. By using the lift we can define the most important concept in this bachelor thesis: the rotation number. We prove that the rotation number is well-defined. We differentiate the rotation number based on its rationality or irrationality. If a homeomorphism f of a given circle has a rational rotation number, then its dynamic is strictly defined; its orbits are either periodic or are converging there in the limit sense. If the rotation number of f is irrational, it is a more complex case. The behaviour of orbits of the homeomorphism f in this case depends on the
degree of smoothness of f.
|
