The embedding flows of \(C^\infty\) hyperbolic diffeomorphisms (Q630559)

In this article, a simple and direct proof of the following statement (Theorem 1.1) is presented: For any \(C^\infty\) locally hyperbolic diffeomorphism \(F(x)= O(|x^2|)\) in \((\mathbb{R}^n,0)\), if \(A\) has a real logarithm \(B\) with its eigenvalues weakly nonresonant, then the diffeomorphism \(F(x)\) admits a \(C^\infty\) embedding flow induced by a \(C^\infty\) vector field of the form \(Bx+ v(x)\) with \(v(x)= O(|x^2|)\). This theorem was mentioned in [\textit{W. Li}, \textit{J. Llibre} and \textit{X. Zhang}, Am. J. Math. 124, No. 1, 107--127 (2002; Zbl 1048.37018)]. It is also shown that the weakly nonresonant condition in the last result on the real logarithm of \(A\) is necessary for some \(C^\infty\) diffeomorphisms \(F(x)= Ax+ f(x)\) to have \(C^\infty\) embedding flows (Theorem 1.2), and it is proved that a germ of \(C^\infty\) hyperbolic diffeomorphisms \(F(x)= Ax+ f(x)\) with \(f(x)= O(|x^2|)\) in \((\mathbb{R}^2, 0)\) has a \(C^\infty\) embedding flow if and only if either \(A\) has no negative eigenvalues or \(A\) has two equal negative eigenvalues and it is diagonalizable (Theorem 1.3).