Dominated splittings for flows with singularities (Q2845642)

Let \(M\) be a connected compact \(n\)-dimensional manifold, \(n\geqslant 3\), with or without boundary, and let \(X:M\to M\) be a vector field. The authors proved the following:NEWLINENEWLINE{ Theorem A}: Let \(\Lambda\) be a compact invariant set of \(X\) such that every singularity in this set is hyperbolic. Suppose that there exists a continuous \(DX_t\)-invariant splitting \(T_{\Lambda}M=E\oplus F\), where \(E\) is uniformly contracting, \(F\) is sectional expanding and assume that for some constants \(C, \lambda>0\) we have \(\|DX_t|_{E_{\sigma}}\|\cdot\|DX_{-t}|_{F_{\sigma}}\|<CE^{-\lambda t}\) for all \(\sigma\in\Lambda\cap\mathrm{Sing}(X)\) and \(t\geq0\). Then \(T_{\Lambda}M=E\oplus F.\)NEWLINENEWLINE{ Theorem B}: Let \(\Lambda\) be a compact invariant set of \(X\) such that every singularity in this set is hyperbolic. Suppose that there exists a continuous \(DX_t\)-invariant splitting \(T_{\Lambda}M=E\oplus F\) such that \(T_{\sigma}M=E_{\sigma}\oplus F_{\sigma}\) is dominated for every singularity \(\sigma\in\Lambda\). If the Lyapunov exponents in the \(E\)-direction are negative and the sectional Lyapunov exponents in the \(F\)-direction are positive on a set with full probability within \(\Lambda\), then the splitting is dominated and \(\Lambda\) is a sectional-hyperbolic set.NEWLINENEWLINE{ Theorem C}: Let \(X\) be a \(C^1\) vector field on a three-dimensional manifold \(M\) admitting a trapping region \(U\) whose singularities are hyperbolic and \(C^1\) linearizable. Let us assume that the compact invariant subset \(\Lambda=\Lambda(U)\) is weakly dissipative and endowed with a one-dimensional continuous field \(F\) of asymptotically backward contracting directions, such that for each \(x\in\Lambda\), \(F_x\) is a one-dimensional subspace of \(T_xM\) and satisfies NEWLINE\[NEWLINE\liminf_{t\to+\infty}\frac{1}{t}\log\|DX_{-t}|_Fx\|<0.NEWLINE\]NEWLINE If at each singularity \(\sigma\in U\) there exists a complementary \(DX_t\)-invariant direction \(E_{\sigma}\) such that \(E_{\sigma}\oplus F_{\sigma}=T_{\sigma}M\) is a dominated splitting, then \(\Lambda\) is a hyperbolic set.