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

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