On robust expansiveness for sectional hyperbolic attracting sets

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).

33 pages; 07 figures; keywords: sectional-hyperbolicity, robust expansiveness, strong dissipativity, star flow, robust transitivity, robust chaotic, attracting sets. Refocused statements of main theorems and proof of more important results. Corrected some typos and improved some definitions

References in corpus (5)

• Entropy theory for sectional hyperbolic flows
• On the statistical stability of families of attracting sets and the contracting Lorenz attractor
• Star flows with singularities of different indices
• Upper, down, two-sided Lorenz attractor, collisions, merging and switching
• Finitely many physical measures for sectional-hyperbolic attracting sets and statistical stability