ABSTRACT. The dynamical degree λ( f ) of a birational transformation f measures the exponential growth rate of the degree of the formulae that define the n-th iterate of f . We study the set of all dynamical degrees of all birational transformations of projective surfaces, and the relationship between the value of λ( f ) and the structure of the conjugacy class of f . For instance, the set of all dynamical degrees of birational transformations of the complex projective plane is a closed and well ordered set of algebraic numbers.

Abstract. We study the algebraic structure of the n-dimensional Cremona group and show that it is not an algebraic group of infinite dimension (indgroup) if n ≥ 2. We describe the obstruction to this, which is of a topological nature.By contrast, we show the existence of a Euclidean topology on the Cremona group which extends that of its classical subgroups and makes it a topological group.

We present in this Note some results on conjugacy classes of finite Abelian subgroups of the Cremona group of the plane

We study the group of birational transformations of the plane that fix (each point of) a curve of geometric genus 1.A precise description of the finite elements is given; it is shown in particular that the order is at most 6, and that if the group contains a non-trivial torsion, the fixed curve is the image of a smooth cubic by a birational transformation of the plane.We show that for a smooth cubic, the group is generated by its elements of degree 3, and prove that it contains a free product of Z/2Z, indexed by the points of the curve.

Abstract. We give the classification of elements -respectively cyclic subgroups -of finite order of the Cremona group, up to conjugation. Natural parametrisations of conjugacy classes, related to fixed curves of positive genus, are provided.Mathematics Subject Classification (2010). 14E07; 20E45; 20G20.