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 .

Arithmetics of the numbers of orbits of the Fermat–Euler dynamical systems

“…Theorem 4 [4] The following holds for the monad graph of the Frobenius map f k : x → x k : a. If G is commutative, then the trees framing the components of the monad f k are isomorphic not only along each component but also along the whole monad graph.…”

“…In [4], some of Arnold's constructions were simplified and several new results on the monad graph of the squaring map (and more generally on the Frobenius monad f k : x → x k ) of a finite group G were obtained. For example, the following was established.…”

“…The respective notation for the squaring monad graph of E(n) and for the doubling monad graph of Z m is [E(n)] and [Z m ]. By Theorem 1 in [4], the doubling graph of Z u with odd u consists of simple cycles, and the doubling graph of Z 2 k is the binary tree T k 2 with k floors. Hence, for an integer M = 2 k u with odd u, the doubling graph of Fig.…”

“…Next, consider the squaring monad of an arbitrary finite group G. It was proved in [4] that (a) the vertices of the cycles of the squaring monad graph are the elements of G that have odd order and (b) the length of any n-cycle of the squaring monad (i.e., of a cycle all of whose elements have order n) is equal to the order of 2 in the Euler group E(n).…”

A converse result concerning the periodic structure of commuting affine circle maps

“…Recall that (Z ∆ , +, ·) is a commutative ring with ∆ elements (+ and · refer the sum and the product of integers modulo ∆, respectively). Moreover, (Z * ∆ , ·) is an Abelian group with Card(Z * ∆ ) = ϕ(∆) (in the literature, it is also called Euler group, see [3,10]). The following well-known result can be found in [2,Theorem 5.17].…”