Topology and statistics of formulae of arithmetics

This paper surveys some recent and classical investigations of geometric progressions of residues that generalize the little Fermat theorem, connect this topic with the theory of dynamical systems, and estimate the degree of chaotic behaviour of systems of residues forming a geometric progression and displaying a distinctive mutual repulsion. As an auxiliary tool, the graphs of squaring operations for the elements of finite groups and rings are studied. For commutative groups the connected components of these graphs turn out to be attracting cycles homogeneously equipped with products of binary rooted trees, the algebra of which is also described in the paper. The equipping with trees turns out to be homogeneous also for the graphs of symmetric groups of permutations, as well as for the groups of even permutations.