DECOMPOSITION OF HOMOGENEOUS VECTOR-FIELDS OF DEGREE ONE AND REPRESENTATION OF THE FLOW

We first give a characterization for the set of real analytic diffeomo rphisms which transform homogeneous vector fields of certain degree in to homogeneous fields of the same degree with respect to an arbitrary dilation delta(epsilon)(tau). Such a set is constituted by the inverti ble analytic maps that are homogeneous of degree one with respect to d elta(epsilon)(tau) and can be endowed with the structure of a Lie Grou p whose Lie algebra is the space H-1,H-tau (R(n)) of the homogeneous f ields of degree one with respect to delta(epsilon)(tau). Then we prove a decomposition theorem for the elements of the non semisimple Lie al gebra H-1,H-tau (R(n)). This result is a non linear analog of the Jord an decomposition of a linear field, i.e. for X is an element of H-1,H- tau (R(n)), we can write X = S + N, with S linear semisimple and [S; N ] = 0. We also give an explicit representation formula for the flow ge nerated by a field in H-1,H-tau (R(n)). Finally we apply this result t o obtain a simple representation for the trajectories of a class of af fine control systems x = X(0)(x) + B u, with X(0) is an element of H-1 ,H-tau (R(n)) and B a constant field, that constitute a natural extens ion of the linear control systems.