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arXiv · 1910.12095

On robust expansiveness for sectional hyperbolic attracting sets

Vitor Araujo·Junilson Cerqueira
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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).

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Vitor Araujo, Junilson Cerqueira. 2023-02-03. On robust expansiveness for sectional hyperbolic attracting sets. https://doi.org/10.17323/1609-4514-2023-23-1-11-46

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