Homoclinic classes for generic C^1 vector fields
C. M. Carballo, C. A. Morales, M. J. Pacifico
TL;DR
The paper extends key structural properties of Axiom A dynamics to generic C^1 flows on closed manifolds by showing that, for a residual set of vector fields, homoclinic classes are neutral, saturated, and maximal transitive, with a robust dependence on periodic data. Central to the approach is the Lyapunov-stability framework and a Hayashi-type connecting lemma, which together yield that homoclinic classes coincide with intersections of Lyapunov-stable sets for X and -X and are isolated exactly when they are Omega-isolated. In three dimensions, the authors derive several equivalent conditions linking hyperbolicity, isolation, and robust transitivity, and establish that finitely many homoclinic classes correspond to a closed union of such classes. Overall, the results demonstrate that generically, homoclinic class structure mimics the classical Axiom A picture without assuming structural stability or Axiom A.
Abstract
We prove that homoclinic classes for a residual set of C^1 vector fields X on closed n-manifolds are maximal transitive and depend continuously on periodic orbit data. In addition, X does not exhibit cycles formed by homoclinic classes. We also prove that a homoclinic class of X is isolated if and only if it is Omega-isolated, and that it is the intersection of its stable set and its unstable set. All these properties are well known for structurally stable Axiom A vector fields.