Geometry and arithmetic of verbal dynamical systems on simple groups
  We study dynamical systems arising from word maps on simple groups. We develop a geometric method based on the classical trace map for investigating periodic points of such systems. These results lead to a new approach to the search of Engel-like sequences of words in two variables which characterize finite solvable groups. They also give rise to some new phenomena and concepts in the arithmetic of dynamical systems. 
Tatiana Bandman, Fritz Grunewald, Boris Kunyavskii, Nathan Jones