The relationship between dynamical systems and operator algebras is one that has been fruitful and mutually beneficial and is by now both well-established, -aged and -matured. This thesis contribute in developing the relationship between dynamical systems, groupoids and operator algebra for circle dynamics. Another aspect of this thesis are the results solely about dynamics on the unit circle.

The reader is supposed to be well versed in C^*-algebras and K-theory (and classification). The reader is not assumed to be familiar with dynamical system theory nor with groupoids.