We show that there exist polynomial endomorphisms of C 2 , possessing a wandering Fatou component. These mappings are polynomial skewproducts, and can be chosen to extend holomorphically of P 2 (C). We also find real examples with wandering domains in R 2 . The proof relies on parabolic implosion techniques, and is based on an original idea of M. Lyubich.

Abstract. We study the dynamics of a bimeromorphic map X → X, where X is a compact complex Kähler surface. Under a natural geometric hypothesis, we construct an invariant probability measure, which is mixing, hyperbolic and of maximal entropy. The proof relies heavily on the theory of laminar currents and is new even in the case of polynomial automorphisms of C 2 . This extends recent results by E. Bedford and J. Diller.

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We study the dynamics of a class of nonalgebraic holomorphic diffeomorphisms, topological analogues in the unit bidisk of complex Hénon mappings in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]. In particular a dynamical degree is defined, which is related to topological entropy, and we construct stable/unstable invariant currents, and prove there is a unique, mixing, measure of maximal entropy, with product structure.

Abstract. We continue our study of the dynamics of meromorphic mappings with small topological degree 2 .f / < 1 .f / on a compact Kähler surface X. Under general hypotheses we are able to construct a canonical invariant measure which is mixing, does not charge pluripolar sets and has a natural geometric description.Our hypotheses are always satisfied when X has Kodaira dimension zero, or when the mapping is induced by a polynomial endomorphism of C 2 . They are new even in the birational case ( 2 .f / D 1). We also exhibit families of mappings where our assumptions are generically satisfied and show that if counterexamples exist, the corresponding measure must give mass to a pluripolar set.Mathematics Subject Classification (2010). 37F10, 32H50, 32U40.

We try to find a geometric interpretation of the wedge product of positive closed laminar currents in C 2 . We say such a wedge product is geometric if it is given by intersecting the disks filling up the currents.Uniformly laminar currents do always intersect geometrically in this sense. We also introduce a class of "strongly approximable" laminar currents, natural from the dynamical point of view, and prove that such currents intersect geometrically provided they have continuous potentials.Date: 24 mars 2003.

We study stability and bifurcations in holomorphic families of polynomial automorphisms of C 2 . We say that such a family is weakly stable over some parameter domain if periodic orbits do not bifurcate there. We first show that this defines a meaningful notion of stability, which parallels in many ways the classical notion of J -stability in one-dimensional dynamics. Define the bifurcation locus to be the complement of the weak stability locus. In the second part of the paper, we prove that under an assumption of moderate dissipativity, the parameters displaying homoclinic tangencies are dense in the bifurcation locus. This confirms one of Palis' Conjectures in the complex setting. The proof relies on the formalism of semi-parabolic bifurcation and the construction of "critical points" in semi-parabolic basins (which makes use of the classical Denjoy-Carleman-Ahlfors and Wiman Theorems).

 J DILLER, R DUJARDIN  V GUEDJ A. -We continue our study of the dynamics of mappings with small topological degree on projective complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic "equilibrium" measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points are equidistributed towards this measure. This generalizes results that were known in the invertible case and adds to the small number of situations in which a natural invariant measure for a non-invertible dynamical system is well-understood.R. -Nous poursuivons notre étude de la dynamique des applications rationnelles de petit degré topologique sur les surfaces complexes projectives. Dans un travail précédent nous avons construit une mesure ergodique naturelle, dite « d'équilibre », sous des hypothèses très générales. Nous étudions maintenant en détail les propriétés dynamiques de cette mesure: nous donnons des bornes optimales pour ses exposants de Lyapounov, montrons qu'elle est d'entropie maximale et qu'elle a une structure produit dans l'extension naturelle. Sous une hypothèse supplémentaire naturelle, nous montrons que cette mesure décrit la répartition des points selles. Ceci généralise des résultats qui étaient auparavant connus dans le cas inversible et vient ainsi s'ajouter au petit nombre de situations où une mesure invariante naturelle pour un système dynamique non inversible est vraiment bien comprise. IntroductionIn this article we continue our investigation, begun in [7,8], of dynamics on complex surfaces for rational transformations with small topological degree. Our previous work culminated in the construction of a canonical mixing invariant measure for a very broad class of such mappings. We intend now to study in detail the nature of this measure. As we will show, J. Diller acknowledges the National Science Foundation for its support through grant DMS 06-53678. R. the measure meets conjectural expectations concerning, among other things, Lyapunov exponents, entropy, product structure in the natural extension, and equidistribution of saddle orbits.Before entering into the details of our results, let us recall our setting. Let X be a complex projective surface (always compact and connected), and f : X → X be a rational mapping. Our main requirement is that f has small topological degree:Here the topological (or second dynamical) degree λ 2 (f ) is the number of preimages of a generic point, whereas the first dynamical degree λ 1 (f ) := lim (f n ) * | H 1,1 (X) 1/n measures the asymptotic volume growth of preimages of curves under iteration of f . We refer the reader to [7] for a more precise discussion of dynamical degrees. In particular it was observed there that the existence of maps with small topological degree imposes some rest...