Dynamique des applications rationnelles des espaces multiprojectifs

We study the dynamics of rational mappings f of ℂ^k by compactifying them in multiprojective spaces ℙⁿ¹ × ⋯ × ℙⁿ⁵. We focus on maps of the surface ℙ¹ × ℙ¹. We follow the approach of [Si 99] and associate to any algebraically stable f an invariant positive closed (1, 1) current. We then consider the existence of an f* invariant measure using the theory of pluripositive currents, and relates it to the measure of Russsakovskii-Shiffman describing the distribution of preimages of points. Our point of view enables us to treat new classes of examples: we consider in particular polynomial skew products with varying degrees, and birational polynomial mappings of ℂ². We also describe the compact convex set of f* invariant currents for monomial and birational maps of ℂ².