Topology of dynamical systems in finite groups and number theory

Abstract

We study and develop a very new object introduced by V.I. Arnold: a monad is a triple consisting of a finite set, a map from that finite set to itself and the monad graph which is the directed graph whose vertices are the elements of the finite set and whose arrows lead each vertex to its image (by the map). We consider the case in which the finite set entering in the monad definition is a finite group G and the map f : G → G is the Frobenius map f k : x ↦ x k , for some k ∈ Z . We study the Frobenius dynamical system defined by the iteration of the monad f k , and also study the combinatorics and topology (i.e., the discrete invariants) of the monad graph. Our study provides useful information about several structures on the group associated to the monad graph. So, for example, several properties of the quadratic residues of finite commutative groups can be obtained in terms of the graph of the Frobenius monad f 2 : x ↦ x 2 .