Invariant Varieties of Periodic Points for the Discrete Euler Top

The Kowalevski workshop on mathematical methods of regular dynamics was organized by Professor Vadim Kuznetsov in April 2000 at the University of Leeds [1]. In his introductory talk about the Kowalevski top, Professor Kuznetzov [2] had shown his strong interest on the subject and motivated the authors to work on classical tops. In our recent paper [3] we have studied the behaviour of periodic points of a rational map and found that they form a variety for each period specified by invariants of the map if the map is integrable, while they form a set of isolated points dependent on the invariants otherwise. It is apparent that an application of our theorem to the problems of a classical top is quite interesting and will be fruitful. We investigate the discrete Euler top, in this article, to see how the invariant varieties of periodic points look like in this particular example. In conclusion we will show that there is no periodic points of period 2 and 4 if the top is not axially symmetric. In the case of period 3 we derive explicitly an algebraic variety of dimension two as an invariant variety of periodic points. When the top is axially symmetric, the angular velocity of a periodic map along the symmetry axis is quantized to some special values determined by the period and the shape of the top. The other components of the angular velocity are free, thus form an invariant variety of periodic points separately for each period.