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arXiv · 2601.01318

A trichotomy for generic sectional-hyperbolic chain-recurrent classes

Elias Rego·Kendry Vivas
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Abstract

The notion of sectional-hyperbolicity is a weakened form of hyperbolicity introduced for vector fields in order to understand the dynamical behavior of certain higher-dimensional systems such as the multidimensional Lorenz attractor. In this paper we address the questions proposed in [\emph{Math. Z.}, \textbf{298} (2021), 469-488] and we provide a partial answer by proving that a $C^1$-generic non-trivial sectional-hyperbolic chain-recurrent class, not necessarily Lyapunov stable, satisfies a trichotomy: it is either a homoclinic loop, a union of saddle connections between singularities, or it is robustly a homoclinic class.

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Elias Rego, Kendry Vivas. 2026-01-04. A trichotomy for generic sectional-hyperbolic chain-recurrent classes. https://arxiv.org/abs/2601.01318

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