On robust expansiveness for sectional hyperbolic attracting sets
Abstract
We prove that sectional-hyperbolic attracting sets for 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 -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 -flow is singular-hyperbolic if, and only if, it is robustly chaotic (robustly sensitive to initial conditions).
Keywords
Cite
@article{arxiv.1910.12095,
title = {On robust expansiveness for sectional hyperbolic attracting sets},
author = {Vitor Araujo and Junilson Cerqueira},
journal= {arXiv preprint arXiv:1910.12095},
year = {2025}
}
Comments
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