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Creation of connections in \(C^1\)-topology. (CrĂ©ation de connexions en topologie \(C^1\).) (French) Zbl 0997.37007
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.
The 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.

MSC:
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37C55 Periodic and quasi-periodic flows and diffeomorphisms
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
58D15 Manifolds of mappings
37C10 Dynamics induced by flows and semiflows
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