Invariant manifolds of maps close to a product of rotations: close to the rotation axis

We consider families of maps depending on a parameter ε such that for ε=0 the map becomes a product of linear rotations in and for ε≠0 the map is weakly attracting in the product of the rotation planes and weakly repelling in some complementary subspace. We prove that the unstable manifold converges to the complementary subspace in the C^r topology, the case r=∞ included. We consider both the local and the global manifolds. For that we prove some results on families of maps near a norm one linear map, which are of independent interest.