Vector Fields with the Oriented Shadowing Property

TL;DR

This paper characterizes the interior of the set of smooth vector fields with the oriented shadowing property, distinguishing structurally stable fields from a special class of non-stable fields, and provides examples.

Contribution

It describes the $ ext{Int}^1( ext{OrientSh})$ set, introduces the class $ ext{B}$ of non-structurally stable vector fields, and shows their relation to the shadowing property.

Findings

The interior of the set of vector fields with the oriented shadowing property coincides with structurally stable fields.

A special class $ ext{B}$ of non-structurally stable vector fields is identified.

An example of a vector field in $ ext{B}$ that belongs to the interior of the shadowing set is provided.

Abstract

We give a description of the $\Cone$-interior ($\Int^1(\OrientSh)$) of the set of smooth vector fields on a smooth closed manifold that have the oriented shadowing property. A special class $\Bb$ of vector fields that are not structurally stable is introduced. It is shown that the set $\Int^1(\OrientSh\setminus\Bb)$ coincides with the set of structurally stable vector fields. An example of a field of the class $\Bb$ belonging to $\Int^1(\OrientSh)$ is given. Bibliography: 18 titles.