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 C1 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.
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