Smooth moduli-free normal forms of hyperbolic germs of diffeomorphisms

In this paper, we study the smooth classifications of germs of diffeomorphisms near a hyperbolic fixed point based on the smooth moduli-free polynomial normal forms of the corresponding diffeomorphisms and give the following result. On, n ≤ 5, with two kinds of exceptions, any two hyperbolic germs of diffeomorphisms with generic nonlinear parts are at least C ¹ conjugated if and only if their linear parts are similar.