Hyperbolicity, stability, and the criterion of homoclinic points.

*(English)*Zbl 0911.58023Let \(M\) be a compact manifold without boundary and let \({\mathfrak X}^1(M)\) be the set of vector fields on \(M\) with \(C^1\) topology. For \(x\in{\mathfrak X}^1(M)\), let \(\omega(x)\) denote the \(\omega\)-limit set of the trajectory \(\gamma_x\) passing through the point \(x\in M\) and similarly \(\alpha(x)\) the \(\alpha\)-limit set of \(\gamma_x\).

In the first section of this survey paper, the following problem is discussed: “For \(p\) and \(q\) respectively belonging to the unstable and stable manifolds of a hyperbolic singularity, if \(\omega(p)\) meets \(\alpha(q)\), then is it possible to have a homoclinic point associated to it by \(C^1\) small perturbation?” Theorems of several authors related to this problem are mentioned. Sections 2 and 3 deal with the history of structural stability and the collapse of the expectation that the stable systems would be dense in the set of all dynamical systems. At the end of the paper, the author announces two theorems related to the solution of the following Palis’ conjecture: “The set of Morse-Smale dynamical systems, together with the systems having a transversal homoclinic point, forms a dense subset in the space of dynamical systems”.

In the first section of this survey paper, the following problem is discussed: “For \(p\) and \(q\) respectively belonging to the unstable and stable manifolds of a hyperbolic singularity, if \(\omega(p)\) meets \(\alpha(q)\), then is it possible to have a homoclinic point associated to it by \(C^1\) small perturbation?” Theorems of several authors related to this problem are mentioned. Sections 2 and 3 deal with the history of structural stability and the collapse of the expectation that the stable systems would be dense in the set of all dynamical systems. At the end of the paper, the author announces two theorems related to the solution of the following Palis’ conjecture: “The set of Morse-Smale dynamical systems, together with the systems having a transversal homoclinic point, forms a dense subset in the space of dynamical systems”.

Reviewer: A.Klíč (Praha)

##### MSC:

37C75 | Stability theory for smooth dynamical systems |

37D99 | Dynamical systems with hyperbolic behavior |

37D15 | Morse-Smale systems |

34D30 | Structural stability and analogous concepts of solutions to ordinary differential equations |