Sectional-Hyperbolicity in Dynamical Systems

• Sectional-hyperbolicity is a dynamical property defined on compact invariant sets with stable contraction and two-dimensional expansion in the central-unstable bundle.
• It generalizes uniform hyperbolicity by requiring sectional (two-dimensional) expansion rather than full multidimensional expansion, making it key for analyzing Lorenz-like attractors.
• The structure features entropy-expansiveness, strong homoclinic classes, and supports SRB measures, providing a framework for statistical and bifurcation studies in higher-dimensional flows.

Sectional-hyperbolicity is a structural property of flows on compact manifolds that extends the framework of uniform hyperbolicity to encompass dynamics where the central direction admits only sectional (i.e., two-dimensional) expansion rather than full multidimensional expansion. This notion plays a central role in the analysis of higher-dimensional singular flows, particularly in the study of attractors such as the geometric Lorenz attractor and its multidimensional analogues. Sectional-hyperbolic sets exhibit a rich and robust dynamical structure, including entropy-expansiveness, a strong homoclinic framework, and statistical properties such as the existence of physical/SRB measures under suitable assumptions.

1. Formal Definition and Canonical Properties

A compact invariant set $\Lambda$ for a $C^1$ vector field $X$ on a compact Riemannian manifold $M$ is sectional-hyperbolic if:

• Hyperbolicity of Singularities: All singularities in $\Lambda$ are hyperbolic, i.e., all eigenvalues of $DX(\sigma)$ have nonzero real parts.
• Invariant Splitting: There exists a continuous, $DX_t$-invariant splitting

$T_\Lambda M = E^s \oplus F$

with $E^s$ (stable) and $F$ (central-unstable), such that - Uniform Contraction on $E^s$: $\|DX_t|_{E^s_x}\| \leq K e^{-\lambda t}$, $\forall x \in \Lambda$, $t \geq 0$. - Domination: $\|DX_t|_{E^s_x}\| < m(DX_t|_{F_x}) e^{-\lambda t}$ where $m(A)$ is the conorm. - Sectional Expansion: $\dim F \geq 2$, and for every $x \in \Lambda$, every two-dimensional subspace $L_x \subset F_x$,

$$|\det(DX_t|_{L_x})| \geq K^{-1}e^{\lambda t}, \qquad \forall t \geq 0.$$

Sectional-hyperbolicity generalizes uniform hyperbolicity by requiring expansion only for two-planes in the center-unstable bundle, not full expansion in all directions. When $\Lambda$ contains no singularities, it is uniformly hyperbolic "of saddle type" (Arbieto et al., 2014, López, 2013, Salgado, 2016, Araujo et al., 2011).

2. Variations, Criteria, and Generalizations

Significant developments include:

• Dominated Splitting Equivalence: The existence of a dominated splitting can, under certain hypotheses, be deduced from sectional expansion at the singularities and contraction of $E^s$ (Araujo et al., 2011, Salgado, 2016).
• Sectional Lyapunov Exponents: For $2 \leq p \leq \dim F$, one defines $p$-sectional Lyapunov exponents; uniform positivity of these exponents for all $p$-planes in $F$ yields $p$-sectional hyperbolicity, which contains classical sectional-hyperbolicity as the $p=2$ case (Salgado, 2016).
• Mostly Nonuniformly Sectional Expanding (MNUSE): This generalization requires only negative time averages of the log inverse two-plane Jacobians on a positive-measure set, strictly containing all previously studied weak hyperbolicity classes such as singular-hyperbolic (SH), asymptotically sectional-hyperbolic (ASH), and multi-singular hyperbolic (MSH) (Araújo et al., 8 Aug 2025).
• Alternative Criteria: Subadditive-cocycle arguments allow the deduction of domination and full sectional-hyperbolicity from contraction plus two-plane expansion, notably by checking domination only at singularities (Araujo et al., 2011).

3. Dynamical Structure: Homoclinic Classes, Entropy, and Statistical Properties

Sectional-hyperbolic sets enforce a strong homoclinic and symbolic dynamics skeleton:

• Homoclinic Structure: Every sectional-hyperbolic Lyapunov-stable set contains a nontrivial homoclinic class (Arbieto et al., 2014).
• Generic Homoclinicity: For $C^1$-generic vector fields, every nontrivial sectional-hyperbolic chain-recurrent class is robustly a homoclinic class, even absent Lyapunov stability (Rego et al., 4 Jan 2026).
• Periodic Orbits and Entropy: Every chain-recurrent (and in particular, Lyapunov-stable) sectional-hyperbolic set has positive topological entropy and contains periodic orbits (Pacifico et al., 2019). Sectional-hyperbolic sets are entropy-expansive, and the entropy varies continuously in the $C^1$ topology.
• SRB/Physical Measures: Sectional-hyperbolic attractors, as well as more general MNUSE attractors, support physical (SRB) measures, unique if the attractor is transitive and under further assumptions on dimension or Lyapunov exponents (Araújo et al., 8 Aug 2025).

4. Bifurcation Theory, Robustness, and Classification

Sectional-hyperbolicity is central in global dynamics and bifurcation theory:

• Palis Conjecture (Dimension 3): For $C^1$-generic flows on compact 3-manifolds, either the flow is singular-hyperbolic (equivalent to sectional-hyperbolic in dimension 3) or it exhibits homoclinic tangencies; singular-hyperbolicity thus describes the $C^1$-dense dynamics away from tangencies (Crovisier et al., 2017).
• Finite Decompositions: Sectional-Anosov flows—those whose maximal invariant set is sectional-hyperbolic—admit a finite decomposition in terms of hyperbolic attractors and Lorenz-like singularities, with their basins dense in phase space (Bautista et al., 2013, López, 2013).
• Finiteness Under Perturbation: There is a uniform upper bound on the number of attractors and repellers arising from small $C^1$ perturbations of a fixed sectional-hyperbolic set (López, 2013, Arbieto et al., 2012).

5. Regularity and Foliation Theory

Sectional-hyperbolic attractors possess regularity properties for their invariant foliations:

• Stable Foliations: For every sectional-hyperbolic attractor, there exists a contracting invariant topological (stable) foliation in a full neighborhood. The smoothness ($C^q$) of this foliation can be verified via explicit "bunching" inequalities depending on spectral properties of the vector field and dominance (Araújo et al., 2016).
• Applications to Lorenz/Lorenz-like Flows: For the classical Lorenz system and perturbations, the stable foliation is $C^{1.278}$, an improvement essential for establishing statistical properties such as exponential mixing.

6. Extension and Limiting Cases

Sectional-hyperbolicity extends the Axiom A paradigm to singular flows but exhibits well-defined limits:

• Intersections and Decomposition: The intersection of a positively and negatively sectional-hyperbolic set (i.e., for the flow and reversed flow) is hyperbolic if both are transitive; in general, the intersection decomposes into hyperbolic, singular, and regular orbit pieces (Bautista et al., 2014).
• Examples Beyond Sectional-Hyperbolicity: Asymptotically sectional-hyperbolic (ASH) sets and MNUSE attractors allow for nonuniform and intermittent expansion, encompassing attractors such as the Rovella attractor which do not satisfy uniform sectional expansion (Rego et al., 2022, Araújo et al., 8 Aug 2025).
• Limits in Classification: There exist "Venice mask" examples and three-dimensional counterexamples showing that sectional-hyperbolicity does not always coincide with homoclinic structure in the nongeneric setting (Bautista et al., 2014, Rego et al., 4 Jan 2026).

7. Connections, Open Problems, and Future Directions

The theory of sectional-hyperbolicity underpins a modern approach to nonuniform and singular hyperbolicity, fostering advances and raising several open problems:

• Higher Codimension and Multi-Singularity Dynamics: Classification and attractor structure for Lyapunov-stable sets in higher codimensions or with multiple singularities remain open (Bautista et al., 2018).
• Symbolic Dynamics and Statistical Properties: Extensions involve symbolic modeling and invariant measure theory for classes with nonuniform expansion (including MNUSE-type attractors) (Araújo et al., 8 Aug 2025, Pacifico et al., 2019).
• Robustness and Parameter Dependence: Sectional-hyperbolicity and associated dynamical phenomena are robust under $C^1$ perturbations, yet finer properties—such as finiteness or the statistical uniqueness of SRB measures—critically depend on domination, expansion rates, and the specific singularity configuration.

These directions continue to connect partial hyperbolicity, singular flows, and modern smooth ergodic theory, establishing sectional-hyperbolicity as a foundational concept in multidimensional, singular, and nonuniformly hyperbolic dynamics (Arbieto et al., 2014, Arbieto et al., 2012, Rego et al., 4 Jan 2026, Araújo et al., 8 Aug 2025, Crovisier et al., 2017, Araújo et al., 2016).