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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 ℙ2. We consider the perturbation (z2 +ɛz +bɛw2, w2) of (z2, w2) and we prove that, for b sufficiently small, it is injective on its basic set Λɛ close to Λ:= {0} × S1. Moreover we give very precise upper and lower estimates for the Hausdorff dimension of the intersection between local stable manifolds and Λɛ, in the case of these maps.