Two theorems concerning variational systems of smooth dynamical systems

V. M. Millionshchikov

Matematičeskie zametki, Tome 5 (1969) no. 1, pp. 49-54.

Résumé

A dynamical system given by a vector field of class $C^2$ in an $n$-dimensional, smooth, closed manifold $V^n$ let us call differentially homogeneous if for every $v,w\in V^n$ there exists a diffeomorphism of $V^n$ 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 fields $f(v)$ 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 $\lambda_1(f)\ge\lambda_2(f)\ge\dots\ge\lambda_n(f)$). It is proven that $\sum_{i=1}^k\lambda_i(f)$ is an upper semicontinuous function of $f(v)$ when $k=1,2,\dots,n$.