Robustly shadowable chain components of \(C^1\) vector fields (Q2870473)

In the study of differentiable dynamical systems, one of the main goals is to understand the structure of the orbits of the systems.NEWLINENEWLINEIn this paper, the authors study the hyperbolic structure on the chain components of \(C^1\) vector fields. Let \(\gamma\) be a hyperbolic closed orbit of a \(C^1\) vector field \(X\) on a closed Riemannian manifold \(M\), and let \(C_X(\gamma)\) be the chain component of \(X\) which contains \(\gamma\). \(C_X(\gamma)\) is said to be \(C^1\) ``robustly shadowable'' if for any \(C^1\) vector field \(Y\) which is \(C^1\) close to \(X\), \(C_Y (\gamma_Y)\) is shadowable for \(Y_t\), where \(\gamma_Y\) denotes the continuation of \(\gamma\) with respect to \(Y\). The authors prove that any \(C^1\) robustly shadowable chain component \(C_X(\gamma)\) does not contain a hyperbolic singularity, and it is hyperbolic if \(C_X(\gamma)\) has no non-hyperbolic singularity.NEWLINENEWLINEThe proof of the main theorem is technically more complicated than in the diffeomorphism case due to the complexity of the calculations which is caused by the time reparametrization in the shadowing theory of vector fields.