Graphs associated with the map X↦X+X−1X↦X+X−1 in finite fields of characteristic three and five

TextIn a previous paper the graphs associated with the iterations of the map ϑ which takes an element x   of a finite field of characteristic two to x+x−1x+x−1 were studied, exploiting the relation between ϑ and the duplication map over Koblitz curves. While in odd characteristic the graphs associated with ϑ seem not to present notable symmetries, these are present in characteristic three and five. In fact, while in characteristic three the map ϑ is conjugated to the inverse of the square mapping, in characteristic five it is related to an endomorphism of a certain elliptic curve. Relying on these considerations we describe the structure of the graphs in finite fields of characteristic three and five and present a computational procedure for constructing examples in any characteristic.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=nnH53jawJaQ.