Birational transformations of projective spaces

Over non-algebraically closed field k there are birational automorphisms, which are regular at every k-point. These automorphisms form a subgroup of the Cremona group, and each such automorphism induce a permutation of points of projective space. Following a paper by Serge Cantat, I will show, which permutations of points of projective space over finite field can be obtained via these birational automorphisms.

More precisely, I will prove, that each permutation of points of projective space $P^n (n \geq 2)$ over fields of odd characteristic and over field of two elements can be obtained from birational automorphism, which is regular at every rational point.