TY - JOUR

T1 - Interiors of sets of vector fields with shadowing corresponding to certain classes of reparameterizations

AU - Tikhomirov, S. B.

PY - 2008/12/1

Y1 - 2008/12/1

N2 - The structure of the C 1-interiors of sets of vector fields with various forms of the shadowing property is studied. The fundamental difference between the problem under consideration and its counterpart for discrete dynamical systems generated by diffeomorphisms is the reparameterization of shadowing orbits. Depending on the type of reparameterization, Lipschitz and oriented shadowing properties are distinguished. As is known, structurally stable vector fields have the Lipschitz shadowing property. Let X be a vector field, and let p and q be its points of rest or closed orbits. Suppose that the stable manifold of p and the unstable manifold of q have a nontransversal intersection point. It is shown that, in this case, the vector field X does not have the Lipschitz shadowing property. If one of the orbits p and q is closed, then X does not have the oriented shadowing property. These assertions imply that the C 1-interior of the set of vector fields with the Lipschitz shadowing property coincides with the set of structurally stable vector fields. If the dimension of the manifold under consideration is at most 3, then a similar result is valid for the oriented shadowing property. We study the structure of the C 1-interiors of sets of vector fields with various forms of the shadowing property. It is shown that, in the case of the Lipschitz shadowing property, it coincides with the set of structurally stable systems. For manifolds of dimension at most 3, a similar result is valid for the oriented shadowing property.

AB - The structure of the C 1-interiors of sets of vector fields with various forms of the shadowing property is studied. The fundamental difference between the problem under consideration and its counterpart for discrete dynamical systems generated by diffeomorphisms is the reparameterization of shadowing orbits. Depending on the type of reparameterization, Lipschitz and oriented shadowing properties are distinguished. As is known, structurally stable vector fields have the Lipschitz shadowing property. Let X be a vector field, and let p and q be its points of rest or closed orbits. Suppose that the stable manifold of p and the unstable manifold of q have a nontransversal intersection point. It is shown that, in this case, the vector field X does not have the Lipschitz shadowing property. If one of the orbits p and q is closed, then X does not have the oriented shadowing property. These assertions imply that the C 1-interior of the set of vector fields with the Lipschitz shadowing property coincides with the set of structurally stable vector fields. If the dimension of the manifold under consideration is at most 3, then a similar result is valid for the oriented shadowing property. We study the structure of the C 1-interiors of sets of vector fields with various forms of the shadowing property. It is shown that, in the case of the Lipschitz shadowing property, it coincides with the set of structurally stable systems. For manifolds of dimension at most 3, a similar result is valid for the oriented shadowing property.

UR - http://www.scopus.com/inward/record.url?scp=80655124193&partnerID=8YFLogxK

U2 - 10.3103/S1063454108040122

DO - 10.3103/S1063454108040122

M3 - Article

AN - SCOPUS:80655124193

VL - 41

SP - 360

EP - 366

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 4

ER -