Theorem of Sternberg-Chen modulo central manifold for Banach spaces

Abstract:
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.