Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory

Diller, Jeffrey; Dujardin, Romain; Guedj, Vincent

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 2, pp. 235-278.

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.

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.

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