This article studies the sequence of iterative degrees of a birational map of the plane. This sequence is known either to be bounded or to have a linear, quadratic or exponential growth.The classification elements of infinite order with a bounded sequence of degrees is achieved, the case of elements of finite order being already known. The coefficients of the linear and quadratic growth are then described, and related to geometrical properties of the map. The dynamical number of base-points is also studied.Applications of our results are the description of embeddings of the Baumslag-Solitar groups and GL(2, Q) into the Cremona group.

Abstract. -A complex compact surface which carries an automorphism of positive topological entropy has been proved by Cantat to be either a torus, a K3 surface, an Enriques surface or a rational surface. Automorphisms of rational surfaces are quite mysterious and have been recently the object of intensive studies. In this paper, we construct several new examples of automorphisms of rational surfaces with positive topological entropy. We also explain how to define and to count parameters in families of birational maps of P 2 (C) and in families of rational surfaces.

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Let k be an uncountable field of characteristic 0. We prove that an automorphism of the group of polynomial automorphisms of k 2 is the composition of an interior automorphism and an automorphism of the field k.

Geometric and dynamic properties of embeddings of SL(2, Z) into the Cremona group are studied. Infinitely many nonconjugate embeddings that preserve the type (i.e., that send elliptic, parabolic and hyperbolic elements onto elements of the same type) are provided. The existence of infinitely many nonconjugate elliptic, parabolic and hyperbolic embeddings is also shown. In particular, a group G of automorphisms of a smooth surface S obtained by blowing up 10 points of the complex projective plane is given. The group G is isomorphic to SL(2, Z), preserves an elliptic curve and all its elements of infinite order are hyperbolic.