Victoria Rayskin (International University Bremen, Germany)

Theorem of Sternberg-Chen modulo central manifold for Banach spaces

We consider C-diffeomorphisms on a Banach space with a fixed point 0. Suppose that these diffeomorphisms have C non-contracting and non-expanding invariant manifolds, and formally conjugate along their intersection (the center). We prove that they admit local C conjugation. In particular, subject to non-resonance condition, there exists a local C linearization of the diffeomorphisms. It also follows that a family of germs with a hyperbolic linear part admits a C linearization, which has C dependence on the parameter of the linearizing family. The results are proved under the assumption that the Banach space allows a special extension of the maps. We discuss corresponding properties of Banach spaces. The proofs of this paper are based on the technique, developed in the works of G. Belitskii.