On the C1 normal forms for hyperbolic vector fields

Given two germs of hyperbolic vector fields associated to autonomous ordinary differential equations x = Ax + ... and y = By + ... where x. y is an element of R-n, and A and B are n x n matrices, we prove that under some algebraic conditions on the eigenvalues of the matrices and genericity condition on the nonlinear terms, they are at least C 1 conjugate if and only if A and B are similar. (C) 2003 Academie des sciences. Published by Editions scientifiques et medicales Elsevier SAS. All rights reserved.