  Based on a group theoretical setting a sort of discrete dynamical system is
constructed and applied to a combinatorial dynamical system defined on the set
of certain Bethe ansatz related objects known as the rigged configurations.
This system is then used to study a one-dimensional periodic cellular automaton
related to discrete Toda lattice. It is shown for the first time that the level
set of this cellular automaton is decomposed into connected components and
every such component is a torus.
