Birational permutations of projective plane (based on works of S.Cantat and S.Asgarli, K.Lai, M.Nakahara and S.Zimmermann)

Let us consider a projective plane over a finite field K. If a birational transformation and its inverse are defined at each K-point, then this transformation defines a permutation of K-points. In the talk we discuss for which fields we can realize any permutation, and for which not.