Elliptic birational automorphisms of the projective space

A birational automorphism $f$ of the projective space $\mathbb{P}^n$ is defined by $n+1$ homogeneous polynomials of the same degree $d$. If these polynomials have no common divisors then $d$ is the degree of the automorphism $f$. Consider the sequence of the degrees of automorphisms $f, f^2, f^3, ...$ The asymptotics of this sequence is a birational invariant of $f$. If the sequence is unbounded then the automorphism has nice dynamical properties which are useful in the study of its geometry. On the other hand, in my talk I am going to discuss the automorphisms $f$ such that degrees of $f^m$ are bounded above by some number; such automorphisms are called elliptic. Blanc and Déserti proved that any elliptic automorphism of $\mathbb{P}^2$ of infinite order is conjugate to a regular automorphism of $\mathbb{P}^2$ in the Cremona group. I am going to tell the proof of this assertion following their paper and then I am going to talk about attempts to generalize this fact to a higher dimension.