American Journal of Mathematics

We study the dynamics of polynomial self mappings f of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]. We construct, for a large class of mappings, an invariant measure μ which is mixing and of maximal entropy h_μ(f) = max (log d_t(f), log λ₁(f)), where d_t(f) is the topological degree of f and λ₁(f) its first dynamical degree. To achieve this, we look at the meromorphic extensions of f to smooth minimal compactifications of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] When a good compactification is found, we construct an f*-invariant Green current T which contains many dynamical informations. When δ := d_t(f)/λ₁(f) > 1, the measure μ is obtained as μ = dd^c(υT), where υ is a partial Green function defined on the support of T. When δ < 1, μ = T ∧ T^- where T^- is a globally defined f_*-invariant current.