On Hölder Dependence of the Parameterized Hartman—Grobman Theorem

The well-known Hartman-Grobman Theorem says that a C1 hyperbolic diffeomorphism F can be locally linearized by a homeomorphism Φ. For parameterized systems Fθ, known results show that the corresponding homeomorphisms Φθ exist uniquely in a functional space equipped with the supremum norm and depend continuously on the parameter θ. In this paper, we further extend the results to Holder dependence of Φθ on θ by Pugh’s strategy, but introducing a kind of special Holder norm instead of the usual supremum norm in the proof to control the linear parts of Fθ. This requires a new Holder linearization result for every Fθ.