On robust expansiveness for sectional hyperbolic attracting sets

TL;DR

The paper proves that sectional-hyperbolic attracting sets are $C^1$-robustly expansive, with codimension-two cases holding unconditionally and higher codimension requiring a $q$-strongly dissipative condition to extend the stable foliation smoothly. The core method is a global Poincaré map on adapted cross-sections, combined with center-unstable cone fields and a quotient map that is expanding on smooth strips, which yields positive expansiveness and, hence, robust expansiveness for nearby vector fields. The work also derives consequences connecting robust expansiveness to star flows, showing that robustly expansive dynamics imply sectional-hyperbolicity under robust transitivity, and establishing a robust chaos–sectional-hyperbolicity correspondence in low dimensions. Collectively, these results strengthen the linkage between expansiveness, chaotic behavior, and hyperbolic-like structure in sectional-hyperbolic systems, with implications for ergodic properties and stability of attractors in $C^1$-open families.

Abstract

We prove that sectional-hyperbolic attracting sets for $C^1$ vector fields are robustly expansive (under an open technical condition of strong dissipative for higher codimensional cases). This extends known results of expansiveness for singular-hyperbolic attractors in $3$-flows even in this low dimensional setting. We deduce some converse results taking advantage of recent progress in the study of star vector fields: a robustly transitive attractor is sectional-hyperbolic if, and only if, it is robustly expansive. In a low dimensional setting, we show that an attracting set of a $3$-flow is singular-hyperbolic if, and only if, it is robustly chaotic (robustly sensitive to initial conditions).

Figures (6)

• Figure 1: Example of a singular-hyperbolic attracting set, non-transitive (in fact, it is the union of two transitive sets indicated by $H_1,H_2$ above) and containing non-Lorenz like singularities.
• Figure 2: Sketch of cross-sections and sub-cross-sections together with the crossing $cu$-disk; and the definition of the Poincaré time at a given point in the $cu$-disk and the corresponding Poincaré map on the stable leaf.
• Figure 3: A sketch of a $cu$-curve connecting two stable leaves on a cross-section with $d_{cu}=2$, on the left; and a possible $cu$-curve connecting two stable leaves on a higher codimensional setting, on the right.
• Figure 4: The neighborhood $V$.
• Figure 5: Sketch of the setting of Lemma \ref{['le:proximo']} on the left; and of the proof of Theorems \ref{['mthm:principal1']} and \ref{['mthm:principal2']} using Theorem \ref{['thm:expansivepoincare']} on the right hand side.

Theorems & Definitions (74)

• Definition 1
• Definition 2
• Theorem A
• Definition 3
• Theorem B
• Remark 2.1
• Conjecture 1
• Theorem 2.2
• Corollary C
• Remark 2.3