Stable sets, hyperbolicity and dimension

In this note we derive an upper bound for the Hausdorff dimension of the stable set of a hyperbolic set $\mathrm{\Lambda }$ of a $C^2$ diffeomorphisms on a $n$-dimensional manifold. As a consequence we obtain that $\dim_H W^s(\Lambda)=n$ is equivalent to the existence of a SRB-measure. We also discuss related results in the case of expanding maps.