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

Abstract

We consider families of maps depending on a parameter epsilon such that for epsilon = 0 the map becomes a product of linear rotations in R2m+n and for epsilon not equal 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 = infinity 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. (C) 2003 Elsevier Science (USA). All rights reserved.