Creation of connections in \(C^1\)-topology (Q2716647)

Consider a class \(C^k\) \((k>0)\) diffeomorphism \(f\) of a manifold \(M\) (not necessarily compact), \(p_0\) a nonperiodic point of \(f\) whose orbit does not tend toward infinity (necessary condition), and two points \(p,q\) belonging to \(M\), the positive (resp. negative) orbit of \(p\) (resp. \(q\)) crossing through a sufficiently small neighborhood of \(p_0\). This paper shows that it is possible to find a perturbation \(g\) of \(f\) in the \(C^1\)-topology such that \(q\) belongs to the positive orbit of \(p\) (basic theorem). Such a result generalizes two previous ones. The first is due S. Hayashi in the case of a compact manifold. The other result was obtained by the author in the simple situation of dimension-two flows. NEWLINENEWLINENEWLINEThe basic theorem permits to connect the two points \(p\) and \(q\), when the intersection (containing a nonperiodic point) of the forward limit set of \(p\) and the backward limit set of \(q\) (generated by the diffeomorphism \(f\)) is not empty. The theorem also allows the creation of very particular connections such as periodic orbits, and connections using pieces of stable, or unstable manifolds of periodic hyperbolic points. Orbit of points, which may be wandering points, can also be closed. Moreover for sufficiently general diffeomorphisms \(f\), the paper shows that some stable sets in the Lyapunov' sense, which are weakly transitive, may be generated. Some of these stable sets are homoclinic classes, therefore truly transitive. The last part of the paper deals with diffeomorphisms preserving the volume, when all the periodic points are hyperbolic.