Distribution of cycle lengths of a quadratic map over finite fields of characteristic 2

The map x 7→ x ² +x defined on a fixed finite field of characteristic 2 is investigated as a dynamical system. The map is known to be a linear map. Its nilpotent points form a subfield, and periodic cycles are somewhat uniform. A general upper bound for the cycle lengths is given in terms of the Carmichael function of the field degree.