ON THE DYNAMICS OF THUE-MORSE SYSTEM OF DIFFERENCE EQUATIONS
Francisco Balibrea (Departamento de Matemáticas, University of Murcia, Spain)
Resumo:
The Thue-Morse system of difference equations was introduced as a model to understand
the electric behavior (conductor or insulator) of an array of electrical punctual positive
charges occupying positions following a one dimensional distribution of points called a
Thue-Morse chain which it is connected to the sequence t = (0110100110010. . . ) called
also the Thue-Morse sequence. Unfolding the system of difference equations, we obtain
the two-dimensional dynamical system in the plane given by
F(x; y) = (x(4 x y); xy) :
The interest of such system was stated by A.Sharkovskii as an open problem and proposing
some questions.
The most interesting dynamics of the system is developed inside an invariant plane
triangle, where hyperbolic periodic points of almost all period appear, there are subsets of
transitivity and invariant curves of spiral form around the unique inside fixed point. We
have proved that the set of periodic orbits is not dense inside the triangle and does not
exist an attractor in Milnor sense.
In this talk we will present also some recent results concerning the behavior of all
points outside the triangle, completing the known dynamics of the system. In fact we
have obtained that outside the triangle, the orbits of all points are unbounded. Some of
them go to infinite in an oscillating way occupying the second and third quadrant of the
plane and others are going in a monotone way to infinite. Outside the triangle there are no
periodic points. Such new results has an interesting interpretation in terms of the physics
of the problem. Additionally we will answer some of the questions stated by Sharkovskii
concerning the inside of the mentioned triangle.
We will also present graphical analysis of the evolution of some orbits and also the
visualization of the dynamics of the system inside the triangle.
Additionally we will present and comment results on another system of difference
equations associate to Fibonacci sequence whose unfolding in R3 is
F(x; y; z) = (y; z; xy z) :
