The graph of a family of functions over quadratic extensions of finite fields

Brochero and Teixeira (2023) [4] showed the behavior of the functional graph of f a ( X ) = X q + 1 + a X 2 over quadratic extensions of finite fields explicitly for a ∈ { 1 , − 1 } . In this article, we create a family of functions using repeated iterations of the function f a ( X ) and taking values of a ∈ { 1 , − 1 } in each iteration. Let α be an n -sequence of values for a , taken over { 1 , − 1 } , and f α ( X ) be the resulting function. We present the form of f α ( X ) and use it to derive a closed formula for the number and length of cycles present in the functional graph of f α ( X ) . We then determine the shape of the trees hanging from each cycle and gather all the results in our main theorem.