Area contraction in the presence of first integrals and almost global convergence (Q998157)

The paper considers a dynamical system in \(\mathbb R^n\) defined by the \(C^1\) vector field \(f\), i.e. by the system of ordinary differential equations \[ \dot{x}=f(x) \] Consider the evolution of the volume of a parallelepiped along the flow defined by the above system, and the evolution of the area of \(k\)-dimensional surfaces with respect to some metric defined by the positive definite matrix \(G\). Further, there are examined the Hausdorff dimension and the evolution of Hausdorff measures, also the box counting dimension. The same subjects are then treated in the presence of \(p\) first integrals defined by the column vector \(h\) satisfying \[ \sum f^i{{\partial h}\over{\partial x^i}}=0 \] and the Jacobian matrix \(\partial h/\partial x\) having full rank everywhere in \(\Omega\subset \mathbb R^n\); the level set \(L_c=\{x:h(x)=c\), \(c\in \mathbb R^p\}\) is invariant and there is considered the restriction of the dynamical system on \(L_c\). The results are illustrated by examples.