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

Graphs associated with the map $x \mapsto x+x^{-1}$ in finite fields of characteristic two