Decomposition of Homogeneous Vector Fields of Degree One and Representation of the Flow

Summary: The paper gives a decomposition theorem for the elements of the nonsemisimple Lie algebra $H^{1,r}(bold R^n)$ of the vector fields on $bold R^n$ that are homogeneous of degree one with respect to a dilation $delta_epsilon^r.$ Each $Xin bold R^n$ is proved to be equal to $S+N,$ with $[S,N]=0$ and $S$ linear semisimple. As a consequence, the author proves that "in absence of esonance" the vector field $X$ is equivalent to its linear part. Finally, the above results are applied to obtain a representation formula for the trajectories of a vector field $X_0in H^{1,r}$ and those of the affine control system $dot x=X_0(x)+Bu$ with $B$ constant of minimum degree.