  We construct the Green current for a random iteration of "horizontal-like"
mappings in two complex dimensions. This is applied to the study of a
polynomial map $f:\mathbb{C}^2\to\mathbb{C}^2$ with the following properties:
  1. infinity is $f$-attracting,
  2. $f$ contracts the line at infinity to a point not in the indeterminacy
set.
  Then the Green current of $f$ can be decomposed into pieces associated with
an itinerary determined by the indeterminacy points.
  We also study the set of escape rates near infinity, i.e. the possible values
of the function $\limsup \frac{1}{n}\log^+\log^+ \norm{f^n}$. We exhibit
examples for which this set contains an interval.
