Abstract:
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.
