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JOURNALS // Matematicheskie Zametki // Archive

Mat. Zametki, 1969 Volume 5, Issue 1, Pages 49–54 (Mi mzm6806)

This article is cited in 1 paper

Two theorems concerning variational systems of smooth dynamical systems

V. M. Millionshchikov

M. V. Lomonosov Moscow State University

Abstract: 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 $k=1,2,\dots,n$.

UDC: 517.9

Received: 16.01.1968


 English version:
Mathematical Notes, 1969, 5:1, 32–35

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