Abstract
We show that if a divergence-free vector field has the C 1 -stably orbital inverse shadowing property with respect to the class of continuous methods T d , then the vector field is Anosov. The results extend the work of Bessa and Rocha (J. Differ. Equ. 250:3960-3966, 2011).MSC:37C10, 37C27, 37C50.
Highlights
The notion of inverse shadowing property is a dual notion of the shadowing property
Lee et al [ ] showed that the C -interior of the set of vector fields with the orbital shadowing property with respect to the class Td coincides with the set of structurally stable vector fields
We study divergence-free vector fields with the inverse, orbital inverse shadowing property with respect to the class Td
Summary
The notion of inverse shadowing property is a dual notion of the shadowing property. It was studied by [ – ]. In [ ], Lee proved that if a diffeomorphism has the C -stably inverse shadowing property with respect to the class of continuous methods Td, the diffeomorphism is structurally stable. Lee [ ] showed that if a volume-preserving diffeomorphism has the C -stably inverse shadowing property with respect to the class Td, the diffeomorphism is Anosov. If a volume-preserving diffeomorphism has the C -stably orbital inverse shadowing property with respect to the class Td, the diffeomorphism is Anosov. In this spirit, we study divergence-free vector fields with the inverse, orbital inverse shadowing property with respect to the class Td. Let us be more detailed.
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