On the hyperbolicity of \(\operatorname{C}^1\)-generic homoclinic classes. (Sur l’hyperbolicité des classes homoclines \(\operatorname{C}^1\)-génériques.) (English. French summary) Zbl 1332.37020
Summary: The works of Liao, Mañé, Franks, Aoki, and Hayashi characterized a lack of hyperbolicity for diffeomorphisms by the existence of weak periodic orbits. In this note, we announce a result that can be seen as a local version of these works: for \(C^1\)-generic diffeomorphisms, a homoclinic class either is hyperbolic or contains a sequence of periodic orbits that have a Lyapunov exponent arbitrarily close to 0.
MSC:
| 37C29 | Homoclinic and heteroclinic orbits for dynamical systems |
| 37D05 | Dynamical systems with hyperbolic orbits and sets |
| 37D25 | Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) |
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