On the global conjugacy of smooth flows (Q2711773)

Let \(X\) and \(Z\) be \(C^k\) complete vector fields on a smooth manifold \(M\). For a differentiable map \(f:M \to M\), denote by \(Tf:TM \to TM\) the induced tangent map. The following integral criterion for global conjugacy of vector fields \(X\) and \(X+Z\) is established: if the integrals \(\text{id}-\int_{0}^{\infty}(T\exp tX)\cdot Z\circ \exp(-t(X+Z)) dt\) and \(\text{id}-\int_{0}^{\infty}(T\exp t( X+Z)\cdot Z\circ \exp(-tX) dt\) converge uniformly on compact subsets of \(M\) respectively to \(C^k (k\geq 1)\) and \(C^1\) mappings \(f\) and \(g\) then \(f\) and \(g\) are \(C^k\) diffeomorphisms of \(M\), \(g=f^{-1}\), and \(f\) conjugates \(X+Z\) with \(X\). The author introduces the notions of decaying and \(C_m\)-decaying of the adjoint flow\((\varphi_t)_*\) generated by \(X\) on a subspace \(E\) of vector fields on \( M\) ~equipped with a nondecreasing countable system of supremum seminorms. In the case where \((\varphi_t)_*\) decays quite simple sufficient conditions are obtained under which the vector field \(X\) in \(\mathbb R^n\) is globally conjugate to \(X+Z\) for sufficiently small \(Z\in E\). The global straightening out theorem for the vector field \(X+Z\) is proved provided that \(Z\) is a \(C^3\)-small fast falling vector field. It is also shown that under the assumption that \(Z\) has globally bounded derivatives and satisfies \(\|Z(x)\|=o(\|x\|)\) the vector field \(-cx+Z, c>0\), is globally \(C^\infty \) conjugate to \(-cx\).NEWLINENEWLINEFor the entire collection see [Zbl 0937.00046].