Dynamical behavior of endomorphisms on certain invariant sets

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 ?². We consider the perturbation (z ² +?z +b?w ², w ²) of (z ², w ²) and we prove that, for b sufficiently small, it is injective on its basic set Λ_? 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 _? , in the case of these maps.