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Dynamics of polynomial mappings of \(\mathbb{C}^2\). (English) Zbl 1198.32007

Summary: We study the dynamics of polynomial self mappings \(f\) of \({\mathbb C} ^2\). We construct, for a large class of mappings, an invariant measure \(\mu\) which is mixing and of maximal entropy \(h_{\mu}(f)=\max (\log d_t(f), \log\lambda _1(f))\), where \(d_t(f)\) is the topological degree of \(f\) and \(\lambda _1(f)\) its first dynamical degree. To achieve this, we look at the meromorphic extensions of \(f\) to smooth minimal compactifications of \({\mathbb C} ^2\). When a good compactification is found, we construct an \(f^*\)-invariant Green current \(T\) which contains many dynamical informations. When \(\delta := d_t(f)/\lambda _1(f)>1\), the measure \(\mu\) is obtained as \(\mu = dd^c(\upsilon T)\), where \(\upsilon\) is a partial Green function defined on the support of \(T\). When \(\delta <1, \mu = T \wedge T^-\) where \(T^-\) is a globally defined \(f_*\)-invariant current.

MSC:

32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
32C30 Integration on analytic sets and spaces, currents
37A25 Ergodicity, mixing, rates of mixing
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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