Holder regularity of Hartman-Grobman theorem on vector bundle

In this paper, we study Holder regularity of the Hartman-Grobman theorem on vector bundles, considering regularity both along the fiber direction and along the base direction.

While the main argument follows standard approaches, additional effort is devoted to constructing a continuous bundle norm that is equivalent to the original norm and with respect to which the linear part becomes hyperbolic at the first iterate.

For the local version of the theorem, we further need to construct a suitable cut-off function on the vector bundle, which is achieved by employing partition of unity.