  In this paper we study the structure of the graphs associated with the
iterations of the map $x \mapsto x+x^{-1}$ over finite fields of characteristic
two. Formulas are given for the length of the cycles and the depth of the trees
relying upon the structure of the group of the rational points of Koblitz
curves and the congruences of Kloosterman sums modulo powers of 2.
