The embedding flows of C∞ hyperbolic diffeomorphisms

In [Weigu Li, J. Llibre, Xiang Zhang, Extension of Floquetʹs theory to nonlinear periodic differential systems and embedding diffeomorphisms in differential flows, Amer. J. Math. 124 (2002) 107–127] we proved that for a germ of C∞ hyperbolic diffeomorphisms F(x)=Ax+f(x) in , if A has a real logarithm with its eigenvalues weakly nonresonant, then F(x) can be embedded in a C∞ autonomous differential system. Its proof was very complicated, which involved the existence of embedding periodic vector field of F(x) and the extension of the Floquetʹs theory to nonlinear C∞ periodic differential systems. In this paper we shall provide a simple and direct proof to this last result. Next we shall show that the weakly nonresonant condition in the last result on the real logarithm of A is necessary for some C∞ diffeomorphisms F(x)=Ax+f(x) to have C∞ embedding flows. Finally we shall prove that a germ of C∞ hyperbolic diffeomorphisms F(x)=Ax+f(x) with f(x)=O(x2) in has a C∞ embedding flow if and only if either A has no negative eigenvalues or A has two equal negative eigenvalues and it can be diagonalizable.