# Moscow Mathematical Journal

Volume 23, Issue 1, January–March 2023  pp. 11–46.

On Robust Expansiveness for Sectional Hyperbolic Attracting Sets

Authors:  Vitor Araujo (1) and Junilson Cerqueira (2)
Author institution:(1) Instituto de Matemática e Estatística, Universidade Federal da Bahia, Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil
(2) Centro de Ciências Exatas e Tecnológicas, Universidade Federal do Recôncavo da Bahia, Rua Rui Barbosa, S/N, 44380-000, Cruz das Almas, Brasil

Summary:

We prove that sectional-hyperbolic attracting sets for $C^1$ vector fields are robustly expansive (under an open technical condition of strong dissipativeness 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 a converse result 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).

2020 Math. Subj. Class. Primary: 37C10; Secondary: 37D30.

Keywords: Sectional-hyperbolicity, robust expansiveness, star flow, strong dissipativity, robust transitivity, robust chaotic, attracting sets.