Interiors of sets of vector fields with shadowing corresponding to certain classes of reparameterizations

TL;DR

This paper investigates the $C^1$-interiors of vector fields with shadowing properties, revealing that for Lipschitz shadowing, the interior coincides with structurally stable fields, and similar results hold in low dimensions for oriented shadowing.

Contribution

It characterizes the $C^1$-interiors of vector fields with shadowing properties, connecting them to structurally stable fields and extending results to low-dimensional cases.

Findings

The $C^1$-interior of Lipschitz shadowing vector fields equals structurally stable fields.

In dimensions up to 3, the $C^1$-interior for oriented shadowing matches that of structurally stable fields.

The results link shadowing properties to structural stability in dynamical systems.

Abstract

We study $C^1$-interiors of sets of vector fields with various shadowing properties. For the case of Lipschitz shadowing property the $C^1$-interior equals the set of structurally stable vector fields. If the dimension of the manifold does not exceed 3 a similar result holds for the oriented shadowing property.