ABSTRACT. We study the Hausdorff dimension of the intersection between local stable manifolds and the respective basic sets of a class of hyperbolic polynomial endomorphisms on the complex projective space P 2 . We consider the perturbation (z 2 + εz + bεw 2 , w 2 ) of (z 2 , w 2 ) and we prove that, for b sufficiently small, it is injective on its basic set Λ ε close to Λ := {0} × S 1 . Moreover we give very precise upper and lower estimates for the Hausdorff dimension of the intersection between local stable manifolds and Λ ε , in the case of these maps.