07-106 V. Rayskin

Theorem of Sternberg-Chen modulo central manifold for Banach spaces (270K, pdf) Apr 29, 07

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Abstract. We consider $C^\infty$-diffeomorphisms on a Banach space with a fixed point $0$. Suppose that these diffeomorphisms have $C^\infty$ non-contracting and non-expanding invariant manifolds, and formally conjugate along their intersection (the center). We prove that they admit local $C^\infty$ conjugation. In particular, subject to non-resonance condition, there exists a local $C^\infty$ linearization of the diffeomorphisms. It also follows that a family of germs with a hyperbolic linear part admits a $C^\infty$ linearization, which has $C^\infty$ 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 (\cite{B1}, \cite{B2}).

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