Arnold's seminar at Moscow University
September 23, 2003

Ricardo Uribe
"On Arnold's Monads of Finite Groups"

We study and developpe a very new object inroduced by V.I. Arnold:
a "monad" is a mapping of a finite set to itself. The "monad's graph"
the graph whose vertices are the elements of the finite set and whose
ORIENTED edges lead each vertex directly to its image (by the monad).

V.I. Arnold discovered that:

        "Every connected component of any monad's graph consists of
         an attracting cycle, framed by rooted trees (whose roots are
         points of the attracting cycle)."

We consider the case in which the finite set entering in monad's
definition is a finite group and we consider the "Frobenius monad"
f_k : x --> x^k, with k integer.

Our study provide information about several structures on the group
associated to the monad's graph of f_k.

For the squring monad, x --> x^2, on a finite group, Arnold proved:

        "Each connected component of the graph of the squaring monad,
         x --> x^2, on a finite group is a cycle, framed homogeneously
         (by isomorphic rooted trees attached to each of its vertices)."

For any Frobenius monad, x --> x^k, on a finite group we prove:

        "The homogeneity of the rooted trees along each connected
         component holds"

        "The vertices of the monad's graph are organised in that graph
         according to their order as elements of that finite group (we
         show explicitly such distribution)"

        "The rooted trees of each connected component of the monad's
         graph are isomorphic to a rooted tree contained in the rooted
         tree attached to the unit element cycle"

For x --> x^k on a COMMUTATIVE finite group G we prove:

        "The trees fraping the components of the maonad's graph are
         isomorphic not only along each connected component but along
         the whole graph"

        "The connected component of the unity element is a subgroup of G
         and also the union of all atracting cycles is a subgroup of G"