Flows with ergodic pseudo orbit tracing property

Abstract

In the manuscript, we deal with a type of pseudo orbit tracing property and hyperbolicity about a vector field (or a divergence free vector field). We prove that a vector field (or a divergence free vector field) of a smooth closed manifold $ M $ has the robustly ergodic pseudo orbit tracing property then it does not contain any singularities and it is Anosov. Additionally, there is a dense and open set $ \mathcal{R} $ in the set of $ C^1 $ a vector field (or a divergence free vector field) of a smooth closed manifold $ M $ such that given a vector field (or a divergence free vector field) has the ergodic pseudo orbit tracing property then it does not contain singularities and it is Anosov.

Keywords

• pseudo orbit tracing property
• ergodic pseudo orbit tracing property
• chain transitive
• transitive
• star condition
• hyperbolic