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

TL;DR

This thesis explores the geometric structure of a discretized Euler top, revealing new birational maps and involutions that describe its dynamics using quadrics, elliptic functions, and geometric transformations.

Contribution

It introduces a geometric framework for the discrete Euler top, identifying new birational maps and involutions that extend understanding of its integrable structure.

Findings

Existence of maps on rulings of quadrics describing the discrete Euler top

Representation of these maps via complex and real involutions

Identification of conditions under which involutions become birational maps

Abstract

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.