On the shadowableness of flows with hyperbolic singularities (Q6624742)

This article presents several results regarding the absence of the shadowing property for a \(C^1\) vector field on a Riemannian manifold.\NThese usually involve an assumption of the existence of a chain-recurrent set containing an attached hyperbolic singularity. The main statements of the paper are:\N\N(1) If a \(C^1\) vector field on a closed and smooth Riemannian manifold has a chain-recurrent set with an attached index-one hyperbolic singularity, then the vector field does not have the shadowing property on this chain-recurrent set;\N\N(2) If a \(C^1\) vector field on a closed and smooth Riemannian manifold of dimension three or less has a chain-recurrent set with an attached hyperbolic singularity, then the vector field does not have the shadowing property on this chain-recurrent set.\N\NAdditional results in the same spirit are provided for noncompact manifolds and rescaled shadowing property.