In this paper x, a are real numbers. Next we state a number of propositions: (1) If a ­ 0 and (x − a) · (x + a) ­ 0, then −a ­ x or x ­ a. (2) If a ¬ 0 and x < a, then x > a. (3) For every point p of E2 T such that |p| ¬ 1 holds −1 ¬ p1 and p1 ¬ 1 and −1 ¬ p2 and p2 ¬ 1. (4) For every point p of E2 T such that |p| ¬ 1 and p1 6= 0 and p2 6= 0 holds −1 < p1 and p1 < 1 and −1 < p2 and p2 < 1. (5) ...

We suggest a geometrical framework in which to discuss periodic layered structures in the unit disk. Bandgaps appear when the point representing the system approaches the unit circle. We show that the trace of the matrix describing the basic period allows for a classification in three families of orbits with quite different properties. The laws of convergence of the iterates to the unit circle ...