Geometry of the discrete time Euler top and related 3-dimensional birational maps

In this thesis we consider a discretization of the Euler top given by Hirota und Kimura. Using the geometric description of the conserved quantities as quadrics in real 3-space, we find that there exist maps on rulings of quadrics in the corresponding pencil, such that the composition of two such maps describes iterations of the discrete time Euler top. We further show that these maps can also be described either by complex involutions, using Jacobi elliptic functions, or by real involutions. Finally, we determine the cases where the real involutions become birational maps.