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In a previous paper the graphs associated with the iterations of the map
? which takes an element
x of a finite field of characteristic two to were studied, exploiting the relation between
? and the duplication map over Koblitz curves. While in odd characteristic the graphs associated with
? seem not to present notable symmetries, these are present in characteristic three and five. In fact, while in characteristic three the map
? 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.
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