## 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 $$f$$.

### MSC:

 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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### References:

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