On the dynamics near infinity of some polynomial mappings in \(\mathbb C^2\). (English) Zbl 1079.37040

Summary: We construct the Green current for a random iteration of horizontal-like mappings in \(\mathbb C^2\). This is applied to the study of a polynomial map \(f : \mathbb C^2 \to \mathbb C^2\) with the following properties: (i) infinity is \(f\)-attracting; (ii) \(f\) contracts the line at infinity to a point not in the indeterminacy set.
We study for such mappings the escape rates near infinity, i.e., the set of possible values of the function \(\lim\sup\frac1n\log^+\log^+\| f^n\|\). We show in particular that the set of possible values can contain an interval.
On the other hand, the Green current \(T\) of \(f\) can be decomposed into pieces associated to an itinerary defined by the indeterminacy points. This allows us to prove that \(\lim\frac1n\log^+\log^+\| f^n\|\) exists \(\| T\|\)-a.e. and we give its value in terms of explicit quantities depending on \(f\).


37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
32U40 Currents
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