Unstable directions and dimension for a class of skew products with overlaps

We study a class of skew products with overlaps in fibers and show that in this case the unstable manifolds really depend on prehistories, even for perturbations of the original maps. We also give several results about the Hausdorff dimension of the fibers of the respective locally maximal invariant set, by using the inverse pressure, the thickness of Cantor sets and some bounds for the preimage counting function.