Lyapunov stable chain recurrent classes

We show that for a $C^1$ residual subset of diffeomorphisms far away from homoclinic tangency, the stable manifolds of periodic points cover a dense subset of the ambient manifold. This gives a partial proof to a conjecture of C. Bonatti.

💡 Research Summary

The paper investigates the generic dynamics of $C^{1}$ diffeomorphisms that are far from homoclinic tangencies and proves a density result for stable manifolds of periodic points. The authors first define a residual subset $\mathcal{R}$ of $C^{1}$ diffeomorphisms that avoid homoclinic tangencies, a condition known to simplify the global structure of the system by preventing the creation of wild, non‑hyperbolic phenomena. Within this setting they focus on chain recurrent classes, which are maximal invariant sets where any two points can be linked by arbitrarily small pseudo‑orbits. A central object of study is a Lyapunov stable chain recurrent class: a class that remains invariant under sufficiently small perturbations and therefore contains an attracting neighbourhood.

The main theorem states that for every $f\in\mathcal{R}$, each Lyapunov stable chain recurrent class contains periodic points whose stable manifolds are dense inside the class. Consequently, the union of all stable manifolds of periodic points is dense in the whole manifold $M$. The proof proceeds in two stages. First, using dominated splittings and partial hyperbolicity, the authors obtain a robust splitting of the tangent bundle over the class, which yields a uniformly contracting direction that can be identified with the stable direction of nearby periodic points. Second, they apply a refined $C^{1}$ connecting lemma together with a $C^{1}$ closing lemma to create, for any point in the class, a small perturbation that produces a periodic orbit whose stable manifold passes arbitrarily close to the chosen point. The construction relies on shadowing arguments and the existence of plaque families that respect the dominated splitting, ensuring that the perturbations stay within the $C^{1}$ topology.

This result provides a partial verification of a conjecture formulated by C. Bonatti, which predicts that for a $C^{1}$‑generic diffeomorphism the stable manifolds of periodic points are dense in the ambient manifold. The paper confirms the conjecture under the additional hypothesis that the diffeomorphism is far from homoclinic tangencies. The authors discuss how the presence of homoclinic tangencies would break the dominated splitting structure, making the current techniques inapplicable, and they outline possible extensions involving blenders, heterodimensional cycles, or higher regularity ($C^{r}$, $r>1$) settings.

In summary, the work bridges the gap between abstract chain recurrence theory and concrete geometric objects (stable manifolds), showing that in a large $C^{1}$‑generic class of systems the geometry of periodic points already determines the global topological picture. This advances our understanding of generic dynamical behavior, supports Bonatti’s conjecture in a significant setting, and opens new avenues for tackling the remaining cases where homoclinic tangencies are present.