Smooth linearization of hyperbolic fixed points without resonance conditions.

*(English)*Zbl 0726.58039The Hartman-Grobman theorem says that near a hyperbolic fixed point a \(C^ 1\)-diffeomorphism in n-dimensional euclidean space is topologically equivalent to its linearization. The author states that for \(C^ 2\)- diffeomorphisms this equivalence is indeed differentiable at the fixed point and Hölder continuous in some neighbourhood of it. For this result no further assumptions are required. In particular, no nonresonance conditions of the Sternberg type are imposed on the eigenvalues of the linear part.

The proof employs the contraction map principle in spaces of functions with suitably chosen norms which incorporate the additional smoothness properties. But unfortunately Proposition 4.1 and its proof are “completely wrong”, as the author admits in a letter to the reviewer, and no correct version has been given yet. Since this proposition is essential for the proof of the basic technical Theorem 3.1, the validity of the main result is in question.

The proof employs the contraction map principle in spaces of functions with suitably chosen norms which incorporate the additional smoothness properties. But unfortunately Proposition 4.1 and its proof are “completely wrong”, as the author admits in a letter to the reviewer, and no correct version has been given yet. Since this proposition is essential for the proof of the basic technical Theorem 3.1, the validity of the main result is in question.

Reviewer: J.Pöschl (Bonn)

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\textit{S. Van Strien}, J. Differ. Equations 85, No. 1, 66--90 (1990; Zbl 0726.58039)

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