  For every $C^2$-small function $B$, we prove that the map $(x,y)\mapsto
(x^2+a,0)+B(x,y,a)$ leaves invariant a physical, SRB probability measure, for a
set of parameters $a$ of positive Lebesgue measure. When the perturbation $B$
is zero, this is the Jakobson Theorem; when the perturbation is a small
constant times $(0,x)$, this is the celebrated Benedicks-Carleson Theorem.
  In particular, a new proof of the last theorem is given, based on
devellopment of the combinatorial formalism of the Yoccoz puzzles. By adding
new geometrical and combinatorial ingredients, and restructuring classic
analytical ideas, we are able to carry out our proof in the $C^2$-topology,
even when the underlying dynamics are given by endomorphisms.
