DIFFERENTIABILITY OF THE CONJUGACY IN THE HARTMAN-GROBMAN THEOREM

The classical Hartman-Grobman Theorem states that a smooth diffeomorphism F(x) near its hyperbolic fixed point x is topological conjugate to its linear part DF((x) over bar) by a local homeomorphism Phi(x). In general, this local homeomorphism is not smooth, not even Lipschitz continuous no matter how smooth F(x) is. A question is: Is this local homeomorphism differentiable at the fixed point? In a 2003 paper by Guysinsky, Hasselblatt and Rayskin, it is shown that for a diffeomorphism F(x), the local homeomorphism indeed is differentiable at the fixed point. In this paper, we prove for a C-1 diffeomorphism F(x) with DF(x) being alpha-Hiilder continuous at the fixed point that the local homeomorphism (I)(x) is differentiable at the fixed point. Here, alpha > 0 depends on the bands of the spectrum of F((x) over bar) for a diffeomorphism in a Banach space. We also give a counterexample showing that the regularity condition on F(x) cannot be lowered to C-1.