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The maps $x \mapsto ax^{2^k}+b$ defined over finite fields of characteristictwo can be related to the duplication map over binary supersingular ellipticcurves. Relying upon the structure of the group of rational points of suchcurves we can describe the possible cycle lengths of the maps. Then we extendour investigation to the maps $x \mapsto (ax^{2^k}+b)^{-1}$. We also noticesome relations between these latter maps and the polynomials $x^{2^k+1} + x+a$, which have been extensively studied in literature.

On the iterations of the maps $ax^{2^k}+b$ and $(a x^{2^k} + b)^{-1}$ over finite fields of characteristic two