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

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