Flows with ergodic pseudo orbit tracing property

Manseob Lee; Manseob Lee

doi:10.3934/era.2022122

Electronic Research Archive

2022, Volume 30, Issue 7: 2406-2416. doi: 10.3934/era.2022122

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Flows with ergodic pseudo orbit tracing property

Manseob Lee ^,

Department of Marketing Big Data and Mathematics, Mokwon University, Daejeon 302729, Korea

Academic Editor: Peter Nandori

Received: 07 November 2021 Revised: 01 April 2022 Accepted: 15 April 2022 Published: 28 April 2022

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.
- pseudo orbit tracing property,
- ergodic pseudo orbit tracing property,
- chain transitive,
- transitive,
- star condition,
- hyperbolic
Citation: Manseob Lee. Flows with ergodic pseudo orbit tracing property[J]. Electronic Research Archive, 2022, 30(7): 2406-2416. doi: 10.3934/era.2022122

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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.

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