Linearization of class C1 for contractions on Banach spaces

AbstractIn this work we prove a C1-linearization result for contraction diffeomorphisms, near a fixed point, valid in infinite-dimensional Banach spaces. As an intermediate step, we prove a specific result of existence of invariant manifolds, which can be interesting by itself and that was needed on the proof of our main theorem. Our results essentially generalize some classical results by P. Hartman in finite dimensions, and a result of Mora–Sola-Morales in the infinite-dimensional case. It is shown that the result can be applied to some abstract systems of semilinear damped wave equations

Known Results and Open Problems on C1 linearization in Banach Spaces

nuloThe purpose of this paper is to review the results obtained by the authors on linearization of dynamical systems in infinite dimensional Banach spaces, especially in the C1 case, and also to present some open problems that we believe that are still important for the understanding of the theory