Two theorems concerning variational systems of smooth dynamical systems

A dynamical system given by a vector field of class C² in an n-dimensional, smooth, closed manifold Vⁿ let us call differentially homogeneous if for every v, w Vⁿ there exists a diffeomorphism of Vⁿ into itself such that it takes v into w and commutes with respect to motion along a trajectory for any time t. It can be shown that all of the variational systems of such a system are almost reducible.Furthermore, the dynamical systems given by the vector fieldsf(v) are considered to be ergodic in that they have the same integral invariant (nearly all of the variational systems of such a system have the same indices ₁(f)₂(f)... _n(f). It is proven that is an upper semicontinuous function off(v) when k=1, 2, ..., n.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 49–54, January, 1969.