Dynamique des applications d'allure polynomiale

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We study the dynamics of polynomial-like mappings in several variables. A special case of our results is the following theorem. Let f be a proper holomorphic map from an open set U onto a Stein manifold V, U⊂⊂ V. Assume f is of topological degree dₜ>1. Then there is a probability measure μsupported on ₙ≥ ₀f⁻ⁿ(V) satisfying the following properties. 1. The measure μis invariant, K-mixing, of maximal entropy dₜ. 2. If J is the Jacobian of f with respect to a volume form then ∫ J μ≥ dₜ. 3. For every probability measure νon V with no mass on pluripolar sets dₜ⁻ⁿ (fⁿ)*ν converges to μ. 4. If the p.s.h. functions on V are μ-integrables (μis PLB) then (a) The Lyapounov exponents for μare strictly positive. (b) μis exponentially mixing. (c) There is a proper analytic subset E of V such that for z∈, μᶻₙ:=dₜ⁻ⁿ (fⁿ)*δz converges to μ. (d) The measure μis a limit of Dirac masses on the repelling periodic points. The condition μis PLB is stable under small pertubation of f. This gives large families where it is satisfied.